Documentation

Mathlib.CategoryTheory.Adjunction.Basic

Adjunctions between functors #

F ⊣ G represents the data of an adjunction between two functors F : C ⥤ D and G : D ⥤ C. F is the left adjoint and G is the right adjoint.

We provide various useful constructors:

There are also typeclasses IsLeftAdjoint / IsRightAdjoint, which asserts the existence of a adjoint functor. Given [F.IsLeftAdjoint], a chosen right adjoint can be obtained as F.rightAdjoint.

Adjunction.comp composes adjunctions.

toEquivalence upgrades an adjunction to an equivalence, given witnesses that the unit and counit are pointwise isomorphisms. Conversely Equivalence.toAdjunction recovers the underlying adjunction from an equivalence.

Overview of the directory CategoryTheory.Adjunction #

structure CategoryTheory.Adjunction {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D C) :
Type (max (max (max u₁ u₂) v₁) v₂)

F ⊣ G represents the data of an adjunction between two functors F : C ⥤ D and G : D ⥤ C. F is the left adjoint and G is the right adjoint.

We use the unit-counit definition of an adjunction. There is a constructor Adjunction.mk' which constructs an adjunction from the data of a hom set equivalence, a unit, and a counit, together with proofs of the equalities homEquiv_unit and homEquiv_counit relating them to each other.

There is also a constructor Adjunction.mkOfHomEquiv which constructs an adjunction from a natural hom set equivalence.

To construct adjoints to a given functor, there are constructors leftAdjointOfEquiv and adjunctionOfEquivLeft (as well as their duals).

See https://stacks.math.columbia.edu/tag/0037.

Instances For
    @[simp]

    Equality of the composition of the unit and counit with the identity F ⟶ FGF ⟶ F = 𝟙

    @[simp]

    Equality of the composition of the unit and counit with the identity G ⟶ GFG ⟶ G = 𝟙

    The notation F ⊣ G stands for Adjunction F G representing that F is left adjoint to G

    Equations
    Instances For

      A class asserting the existence of a right adjoint.

      Instances
        theorem CategoryTheory.Functor.IsLeftAdjoint.exists_rightAdjoint {C : Type u₁} :
        ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {D : Type u₂} {inst_1 : CategoryTheory.Category.{v₂, u₂} D} {left : CategoryTheory.Functor C D} [self : left.IsLeftAdjoint], ∃ (right : CategoryTheory.Functor D C), Nonempty (left right)

        A class asserting the existence of a left adjoint.

        Instances
          theorem CategoryTheory.Functor.IsRightAdjoint.exists_leftAdjoint {C : Type u₁} :
          ∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {D : Type u₂} {inst_1 : CategoryTheory.Category.{v₂, u₂} D} {right : CategoryTheory.Functor D C} [self : right.IsRightAdjoint], ∃ (left : CategoryTheory.Functor C D), Nonempty (left right)

          A chosen left adjoint to a functor that is a right adjoint.

          Equations
          • R.leftAdjoint = .choose
          Instances For

            A chosen right adjoint to a functor that is a left adjoint.

            Equations
            • L.rightAdjoint = .choose
            Instances For
              noncomputable def CategoryTheory.Adjunction.ofIsLeftAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (left : CategoryTheory.Functor C D) [left.IsLeftAdjoint] :
              left left.rightAdjoint

              The adjunction associated to a functor known to be a left adjoint.

              Equations
              Instances For
                noncomputable def CategoryTheory.Adjunction.ofIsRightAdjoint {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (right : CategoryTheory.Functor C D) [right.IsRightAdjoint] :
                right.leftAdjoint right

                The adjunction associated to a functor known to be a right adjoint.

                Equations
                Instances For
                  @[simp]
                  theorem CategoryTheory.Adjunction.left_triangle_components_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : F G) (X : C) {Z : D} (h : F.obj X Z) :
                  CategoryTheory.CategoryStruct.comp (F.map (self.unit.app X)) (CategoryTheory.CategoryStruct.comp (self.counit.app (F.obj X)) h) = h
                  @[simp]
                  theorem CategoryTheory.Adjunction.right_triangle_components_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : F G) (Y : D) {Z : C} (h : G.obj Y Z) :
                  CategoryTheory.CategoryStruct.comp (self.unit.app (G.obj Y)) (CategoryTheory.CategoryStruct.comp (G.map (self.counit.app Y)) h) = h
                  theorem CategoryTheory.Adjunction.homEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (f : F.obj X Y) :
                  (adj.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map f)
                  theorem CategoryTheory.Adjunction.homEquiv_symm_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (g : X G.obj Y) :
                  (adj.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app Y)
                  def CategoryTheory.Adjunction.homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) :
                  (F.obj X Y) (X G.obj Y)

                  The hom set equivalence associated to an adjunction.

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For
                    theorem CategoryTheory.Adjunction.homEquiv_unit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (f : F.obj X Y) :
                    (adj.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map f)

                    Alias of CategoryTheory.Adjunction.homEquiv_apply.

                    theorem CategoryTheory.Adjunction.homEquiv_counit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) (Y : D) (g : X G.obj Y) :
                    (adj.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app Y)

                    Alias of CategoryTheory.Adjunction.homEquiv_symm_apply.

                    Equations
                    • =
                    Equations
                    • =
                    theorem CategoryTheory.Adjunction.homEquiv_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) :
                    (adj.homEquiv X (F.obj X)) (CategoryTheory.CategoryStruct.id (F.obj X)) = adj.unit.app X
                    theorem CategoryTheory.Adjunction.homEquiv_symm_id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) :
                    (adj.homEquiv (G.obj X) X).symm (CategoryTheory.CategoryStruct.id (G.obj X)) = adj.counit.app X
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : X G.obj Y) :
                    (adj.homEquiv X' Y).symm (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (F.map f) ((adj.homEquiv X Y).symm g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : F.obj X Y) :
                    (adj.homEquiv X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((adj.homEquiv X Y) g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : D} {Y' : D} (f : F.obj X Y) (g : Y Y') :
                    (adj.homEquiv X Y') (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X Y) f) (G.map g)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : D} {Y' : D} (f : X G.obj Y) (g : Y Y') :
                    (adj.homEquiv X Y').symm (CategoryTheory.CategoryStruct.comp f (G.map g)) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X Y).symm f) g
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp (F.map f) g = CategoryTheory.CategoryStruct.comp h✝ k) {Z : C} (h : G.obj Y' Z) :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp (F.map f) g = CategoryTheory.CategoryStruct.comp h k) :
                    CategoryTheory.CategoryStruct.comp f ((adj.homEquiv X Y') g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X' Y) h) (G.map k)
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp h✝ (G.map k)) {Z : D} (h : Y' Z) :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') (w : CategoryTheory.CategoryStruct.comp f g = CategoryTheory.CategoryStruct.comp h (G.map k)) :
                    CategoryTheory.CategoryStruct.comp (F.map f) ((adj.homEquiv X Y').symm g) = CategoryTheory.CategoryStruct.comp ((adj.homEquiv X' Y).symm h) k
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_left_square_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : F.obj X Y') (h : F.obj X' Y) (k : Y Y') :
                    theorem CategoryTheory.Adjunction.homEquiv_naturality_right_square_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X' : C} {X : C} {Y : D} {Y' : D} (f : X' X) (g : X G.obj Y') (h : X' G.obj Y) (k : Y Y') :
                    @[simp]
                    theorem CategoryTheory.Adjunction.counit_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : D} {Y : D} (f : X Y) :
                    CategoryTheory.CategoryStruct.comp (F.map (G.map f)) (adj.counit.app Y) = CategoryTheory.CategoryStruct.comp (adj.counit.app X) f
                    @[simp]
                    theorem CategoryTheory.Adjunction.unit_naturality {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {X : C} {Y : C} (f : X Y) :
                    CategoryTheory.CategoryStruct.comp (adj.unit.app X) (G.map (F.map f)) = CategoryTheory.CategoryStruct.comp f (adj.unit.app Y)
                    theorem CategoryTheory.Adjunction.unit_comp_map_eq_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    CategoryTheory.CategoryStruct.comp (adj.unit.app A) (G.map f) = g f = CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app B)
                    theorem CategoryTheory.Adjunction.eq_unit_comp_map_iff {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    g = CategoryTheory.CategoryStruct.comp (adj.unit.app A) (G.map f) CategoryTheory.CategoryStruct.comp (F.map g) (adj.counit.app B) = f
                    theorem CategoryTheory.Adjunction.homEquiv_apply_eq {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    (adj.homEquiv A B) f = g f = (adj.homEquiv A B).symm g
                    theorem CategoryTheory.Adjunction.eq_homEquiv_apply {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) {A : C} {B : D} (f : F.obj A B) (g : A G.obj B) :
                    g = (adj.homEquiv A B) f (adj.homEquiv A B).symm g = f
                    @[simp]
                    theorem CategoryTheory.Adjunction.corepresentableBy_homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) :
                    ∀ {Y : D}, (adj.corepresentableBy X).homEquiv = adj.homEquiv X Y
                    def CategoryTheory.Adjunction.corepresentableBy {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : C) :
                    (G.comp (CategoryTheory.coyoneda.obj (Opposite.op X))).CorepresentableBy (F.obj X)

                    If adj : F ⊣ G, and X : C, then F.obj X corepresents Y ↦ (X ⟶ G.obj Y)

                    Equations
                    • adj.corepresentableBy X = { homEquiv := fun {Y : D} => adj.homEquiv X Y, homEquiv_comp := }
                    Instances For
                      @[simp]
                      theorem CategoryTheory.Adjunction.representableBy_homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (Y : D) :
                      ∀ {X : C}, (adj.representableBy Y).homEquiv = (adj.homEquiv X Y).symm
                      def CategoryTheory.Adjunction.representableBy {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (Y : D) :
                      (F.op.comp (CategoryTheory.yoneda.obj Y)).RepresentableBy (G.obj Y)

                      If adj : F ⊣ G, and Y : D, then G.obj Y represents X ↦ (F.obj X ⟶ Y)

                      Equations
                      • adj.representableBy Y = { homEquiv := fun {X : C} => (adj.homEquiv X Y).symm, homEquiv_comp := }
                      Instances For

                        This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mk'. This structure won't typically be used anywhere else.

                        • homEquiv : (X : C) → (Y : D) → (F.obj X Y) (X G.obj Y)

                          The equivalence between Hom (F X) Y and Hom X (G Y) coming from an adjunction

                        • unit : CategoryTheory.Functor.id C F.comp G

                          The unit of an adjunction

                        • counit : G.comp F CategoryTheory.Functor.id D

                          The counit of an adjunction

                        • homEquiv_unit : ∀ {X : C} {Y : D} {f : F.obj X Y}, (self.homEquiv X Y) f = CategoryTheory.CategoryStruct.comp (self.unit.app X) (G.map f)

                          The relationship between the unit and hom set equivalence of an adjunction

                        • homEquiv_counit : ∀ {X : C} {Y : D} {g : X G.obj Y}, (self.homEquiv X Y).symm g = CategoryTheory.CategoryStruct.comp (F.map g) (self.counit.app Y)

                          The relationship between the counit and hom set equivalence of an adjunction

                        Instances For

                          The relationship between the unit and hom set equivalence of an adjunction

                          The relationship between the counit and hom set equivalence of an adjunction

                          This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mkOfHomEquiv. This structure won't typically be used anywhere else.

                          Instances For
                            @[simp]
                            theorem CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left_symm {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (self : CategoryTheory.Adjunction.CoreHomEquiv F G) {X' : C} {X : C} {Y : D} (f : X' X) (g : X G.obj Y) :
                            (self.homEquiv X' Y).symm (CategoryTheory.CategoryStruct.comp f g) = CategoryTheory.CategoryStruct.comp (F.map f) ((self.homEquiv X Y).symm g)

                            The property that describes how homEquiv.symm transforms compositions X' ⟶ X ⟶ G Y

                            @[simp]

                            The property that describes how homEquiv transforms compositions F X ⟶ Y ⟶ Y'

                            This is an auxiliary data structure useful for constructing adjunctions. See Adjunction.mkOfUnitCounit. This structure won't typically be used anywhere else.

                            Instances For
                              @[simp]

                              Equality of the composition of the unit, associator, and counit with the identity F ⟶ (F G) F ⟶ F (G F) ⟶ F = NatTrans.id F

                              @[simp]

                              Equality of the composition of the unit, associator, and counit with the identity G ⟶ G (F G) ⟶ (F G) F ⟶ G = NatTrans.id G

                              Construct an adjunction from the data of a CoreHomEquivUnitCounit, i.e. a hom set equivalence, unit and counit natural transformations together with proofs of the equalities homEquiv_unit and homEquiv_counit relating them to each other.

                              Equations
                              Instances For

                                Construct an adjunction between F and G out of a natural bijection between each F.obj X ⟶ Y and X ⟶ G.obj Y.

                                Equations
                                • One or more equations did not get rendered due to their size.
                                Instances For

                                  Construct an adjunction between functors F and G given a unit and counit for the adjunction satisfying the triangle identities.

                                  Equations
                                  Instances For

                                    The adjunction between the identity functor on a category and itself.

                                    Equations
                                    • One or more equations did not get rendered due to their size.
                                    Instances For
                                      Equations
                                      • CategoryTheory.Adjunction.instInhabitedId = { default := CategoryTheory.Adjunction.id }

                                      If F and G are naturally isomorphic functors, establish an equivalence of hom-sets.

                                      Equations
                                      • One or more equations did not get rendered due to their size.
                                      Instances For

                                        If G and H are naturally isomorphic functors, establish an equivalence of hom-sets.

                                        Equations
                                        • One or more equations did not get rendered due to their size.
                                        Instances For

                                          Transport an adjunction along a natural isomorphism on the left.

                                          Equations
                                          • One or more equations did not get rendered due to their size.
                                          Instances For

                                            Transport an adjunction along a natural isomorphism on the right.

                                            Equations
                                            • One or more equations did not get rendered due to their size.
                                            Instances For
                                              @[simp]
                                              theorem CategoryTheory.Adjunction.compYonedaIso_inv_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) (X : Cᵒᵖ) :
                                              ∀ (a : ((CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type v₁)).obj F.op)).obj X✝).obj X), (adj.compYonedaIso.inv.app X✝).app X a = CategoryTheory.CategoryStruct.comp (adj.unit.app (Opposite.unop X)) (G.map a)
                                              @[simp]
                                              theorem CategoryTheory.Adjunction.compYonedaIso_hom_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : D) (X : Cᵒᵖ) :
                                              ∀ (a : ((G.comp CategoryTheory.yoneda).obj X✝).obj X), (adj.compYonedaIso.hom.app X✝).app X a = CategoryTheory.CategoryStruct.comp (F.map a) (adj.counit.app X✝)
                                              def CategoryTheory.Adjunction.compYonedaIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) :
                                              G.comp CategoryTheory.yoneda CategoryTheory.yoneda.comp ((CategoryTheory.whiskeringLeft Cᵒᵖ Dᵒᵖ (Type v₁)).obj F.op)

                                              The isomorpism which an adjunction F ⊣ G induces on G ⋙ yoneda. This states that Adjunction.homEquiv is natural in both arguments.

                                              Equations
                                              Instances For
                                                @[simp]
                                                theorem CategoryTheory.Adjunction.compCoyonedaIso_inv_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : Cᵒᵖ) (X : D) :
                                                ∀ (a : ((CategoryTheory.coyoneda.comp ((CategoryTheory.whiskeringLeft D C (Type v₁)).obj G)).obj X✝).obj X), (adj.compCoyonedaIso.inv.app X✝).app X a = CategoryTheory.CategoryStruct.comp (F.map a) (adj.counit.app X)
                                                @[simp]
                                                theorem CategoryTheory.Adjunction.compCoyonedaIso_hom_app_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) (X : Cᵒᵖ) (X : D) :
                                                ∀ (a : ((F.op.comp CategoryTheory.coyoneda).obj X✝).obj X), (adj.compCoyonedaIso.hom.app X✝).app X a = CategoryTheory.CategoryStruct.comp (adj.unit.app (Opposite.unop X✝)) (G.map a)
                                                def CategoryTheory.Adjunction.compCoyonedaIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₁, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) :
                                                F.op.comp CategoryTheory.coyoneda CategoryTheory.coyoneda.comp ((CategoryTheory.whiskeringLeft D C (Type v₁)).obj G)

                                                The isomorpism which an adjunction F ⊣ G induces on F.op ⋙ coyoneda. This states that Adjunction.homEquiv is natural in both arguments.

                                                Equations
                                                Instances For

                                                  Composition of adjunctions.

                                                  See https://stacks.math.columbia.edu/tag/0DV0.

                                                  Equations
                                                  • One or more equations did not get rendered due to their size.
                                                  Instances For
                                                    theorem CategoryTheory.Adjunction.comp_unit_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : C) {Z : C} (h : G.obj (I.obj (H.obj (F.obj X))) Z) :
                                                    CategoryTheory.CategoryStruct.comp ((adj₁.comp adj₂).unit.app X) h = CategoryTheory.CategoryStruct.comp (adj₁.unit.app X) (CategoryTheory.CategoryStruct.comp (G.map (adj₂.unit.app (F.obj X))) h)
                                                    @[simp]
                                                    theorem CategoryTheory.Adjunction.comp_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : C) :
                                                    (adj₁.comp adj₂).unit.app X = CategoryTheory.CategoryStruct.comp (adj₁.unit.app X) (G.map (adj₂.unit.app (F.obj X)))
                                                    theorem CategoryTheory.Adjunction.comp_counit_app_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : E) {Z : E} (h : X Z) :
                                                    CategoryTheory.CategoryStruct.comp ((adj₁.comp adj₂).counit.app X) h = CategoryTheory.CategoryStruct.comp (H.map (adj₁.counit.app (I.obj X))) (CategoryTheory.CategoryStruct.comp (adj₂.counit.app X) h)
                                                    @[simp]
                                                    theorem CategoryTheory.Adjunction.comp_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) (X : E) :
                                                    (adj₁.comp adj₂).counit.app X = CategoryTheory.CategoryStruct.comp (H.map (adj₁.counit.app (I.obj X))) (adj₂.counit.app X)
                                                    theorem CategoryTheory.Adjunction.comp_homEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} {E : Type u₃} [ℰ : CategoryTheory.Category.{v₃, u₃} E] {H : CategoryTheory.Functor D E} {I : CategoryTheory.Functor E D} (adj₁ : F G) (adj₂ : H I) :
                                                    (adj₁.comp adj₂).homEquiv = fun (x : C) (x_1 : E) => (adj₂.homEquiv (F.obj x) x_1).trans (adj₁.homEquiv x (I.obj x_1))
                                                    @[simp]
                                                    theorem CategoryTheory.Adjunction.leftAdjointOfEquiv_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :
                                                    ∀ (a : C), (CategoryTheory.Adjunction.leftAdjointOfEquiv e he).obj a = F_obj a
                                                    @[simp]
                                                    theorem CategoryTheory.Adjunction.leftAdjointOfEquiv_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) {X : C} {X' : C} (f : X X') :
                                                    (CategoryTheory.Adjunction.leftAdjointOfEquiv e he).map f = (e X (F_obj X')).symm (CategoryTheory.CategoryStruct.comp f ((e X' (F_obj X')) (CategoryTheory.CategoryStruct.id (F_obj X'))))
                                                    def CategoryTheory.Adjunction.leftAdjointOfEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :

                                                    Construct a left adjoint functor to G, given the functor's value on objects F_obj and a bijection e between F_obj X ⟶ Y and X ⟶ G.obj Y satisfying a naturality law he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g. Dual to rightAdjointOfEquiv.

                                                    Equations
                                                    • One or more equations did not get rendered due to their size.
                                                    Instances For
                                                      @[simp]
                                                      theorem CategoryTheory.Adjunction.adjunctionOfEquivLeft_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) (Y : D) :
                                                      (CategoryTheory.Adjunction.adjunctionOfEquivLeft e he).counit.app Y = (e (G.obj Y) Y).symm (CategoryTheory.CategoryStruct.id (G.obj Y))
                                                      @[simp]
                                                      theorem CategoryTheory.Adjunction.adjunctionOfEquivLeft_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) (X : C) :
                                                      def CategoryTheory.Adjunction.adjunctionOfEquivLeft {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {G : CategoryTheory.Functor D C} {F_obj : CD} (e : (X : C) → (Y : D) → (F_obj X Y) (X G.obj Y)) (he : ∀ (X : C) (Y Y' : D) (g : Y Y') (h : F_obj X Y), (e X Y') (CategoryTheory.CategoryStruct.comp h g) = CategoryTheory.CategoryStruct.comp ((e X Y) h) (G.map g)) :

                                                      Show that the functor given by leftAdjointOfEquiv is indeed left adjoint to G. Dual to adjunctionOfRightEquiv.

                                                      Equations
                                                      Instances For
                                                        @[simp]
                                                        theorem CategoryTheory.Adjunction.rightAdjointOfEquiv_map {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) {Y : D} {Y' : D} (g : Y Y') :
                                                        @[simp]
                                                        theorem CategoryTheory.Adjunction.rightAdjointOfEquiv_obj {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :
                                                        ∀ (a : D), (CategoryTheory.Adjunction.rightAdjointOfEquiv e he).obj a = G_obj a
                                                        def CategoryTheory.Adjunction.rightAdjointOfEquiv {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :

                                                        Construct a right adjoint functor to F, given the functor's value on objects G_obj and a bijection e between F.obj X ⟶ Y and X ⟶ G_obj Y satisfying a naturality law he : ∀ X Y Y' g h, e X' Y (F.map f ≫ g) = f ≫ e X Y g. Dual to leftAdjointOfEquiv.

                                                        Equations
                                                        • One or more equations did not get rendered due to their size.
                                                        Instances For
                                                          @[simp]
                                                          theorem CategoryTheory.Adjunction.adjunctionOfEquivRight_counit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) (Y : D) :
                                                          (CategoryTheory.Adjunction.adjunctionOfEquivRight e he).counit.app Y = (e (G_obj Y) Y).symm (CategoryTheory.CategoryStruct.id (G_obj Y))
                                                          @[simp]
                                                          theorem CategoryTheory.Adjunction.adjunctionOfEquivRight_unit_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) (X : C) :
                                                          def CategoryTheory.Adjunction.adjunctionOfEquivRight {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G_obj : DC} (e : (X : C) → (Y : D) → (F.obj X Y) (X G_obj Y)) (he : ∀ (X' X : C) (Y : D) (f : X' X) (g : F.obj X Y), (e X' Y) (CategoryTheory.CategoryStruct.comp (F.map f) g) = CategoryTheory.CategoryStruct.comp f ((e X Y) g)) :

                                                          Show that the functor given by rightAdjointOfEquiv is indeed right adjoint to F. Dual to adjunctionOfEquivRight.

                                                          Equations
                                                          Instances For
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_counitIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : D) :
                                                            adj.toEquivalence.counitIso.hom.app X = adj.counit.app X
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_unitIso_hom_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : C) :
                                                            adj.toEquivalence.unitIso.hom.app X = adj.unit.app X
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_unitIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : C) :
                                                            adj.toEquivalence.unitIso.inv.app X = CategoryTheory.inv (adj.unit.app X)
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_inverse {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                            adj.toEquivalence.inverse = G
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_functor {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                            adj.toEquivalence.functor = F
                                                            @[simp]
                                                            theorem CategoryTheory.Adjunction.toEquivalence_counitIso_inv_app {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] (X : D) :
                                                            adj.toEquivalence.counitIso.inv.app X = CategoryTheory.inv (adj.counit.app X)
                                                            noncomputable def CategoryTheory.Adjunction.toEquivalence {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C} (adj : F G) [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] :
                                                            C D

                                                            If the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the adjunction to an equivalence.

                                                            Equations
                                                            • One or more equations did not get rendered due to their size.
                                                            Instances For

                                                              If the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise) isomorphisms, then the functor is an equivalence of categories.

                                                              @[simp]
                                                              @[simp]

                                                              The adjunction given by an equivalence of categories. (To obtain the opposite adjunction, simply use e.symm.toAdjunction.

                                                              Equations
                                                              • e.toAdjunction = { unit := e.unit, counit := e.counit, left_triangle_components := , right_triangle_components := }
                                                              Instances For
                                                                instance CategoryTheory.Functor.isLeftAdjoint_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) [F.IsLeftAdjoint] [G.IsLeftAdjoint] :
                                                                (F.comp G).IsLeftAdjoint

                                                                If F and G are left adjoints then F ⋙ G is a left adjoint too.

                                                                Equations
                                                                • =
                                                                instance CategoryTheory.Functor.isRightAdjoint_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D E} [F.IsRightAdjoint] [G.IsRightAdjoint] :
                                                                (F.comp G).IsRightAdjoint

                                                                If F and G are right adjoints then F ⋙ G is a right adjoint too.

                                                                Equations
                                                                • =

                                                                Transport being a right adjoint along a natural isomorphism.

                                                                Transport being a left adjoint along a natural isomorphism.

                                                                An equivalence E is left adjoint to its inverse.

                                                                Equations
                                                                • E.adjunction = E.asEquivalence.toAdjunction
                                                                Instances For
                                                                  @[instance 10]

                                                                  If F is an equivalence, it's a left adjoint.

                                                                  Equations
                                                                  • =
                                                                  @[instance 10]

                                                                  If F is an equivalence, it's a right adjoint.

                                                                  Equations
                                                                  • =