# Adjoints of fully faithful functors #

A left adjoint is fully faithful, if and only if the unit is an isomorphism (and similarly for right adjoints and the counit).

adjunction.restrict_fully_faithful shows that an adjunction can be restricted along fully faithful inclusions.

## Future work #

The statements from Riehl 4.5.13 for adjoints which are either full, or faithful.

@[protected, instance]
def category_theory.unit_is_iso_of_L_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the left adjoint is fully faithful, then the unit is an isomorphism.

See

@[protected, instance]
def category_theory.counit_is_iso_of_R_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the right adjoint is fully faithful, then the counit is an isomorphism.

See https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)

@[simp]
theorem category_theory.inv_map_unit {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) {X : C} [category_theory.is_iso (h.unit.app X)] :

If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given by L whiskered with the counit.

@[simp]
theorem category_theory.whisker_left_L_counit_iso_of_is_iso_unit_hom_app {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) (X : C) :
@[simp]
theorem category_theory.whisker_left_L_counit_iso_of_is_iso_unit_inv_app {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) (X : C) :
noncomputable def category_theory.whisker_left_L_counit_iso_of_is_iso_unit {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
L R L L

If the unit is an isomorphism, bundle one has an isomorphism L ⋙ R ⋙ L ≅ L.

Equations
@[simp]
theorem category_theory.inv_counit_map {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) {X : D} [category_theory.is_iso (h.counit.app X)] :

If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given by R whiskered with the unit.

@[simp]
theorem category_theory.whisker_left_R_unit_iso_of_is_iso_counit_hom_app {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) (X : D) :
noncomputable def category_theory.whisker_left_R_unit_iso_of_is_iso_counit {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
R L R R

If the counit of an is an isomorphism, one has an isomorphism (R ⋙ L ⋙ R) ≅ R.

Equations
@[simp]
theorem category_theory.whisker_left_R_unit_iso_of_is_iso_counit_inv_app {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R) (X : D) :
noncomputable def category_theory.L_full_of_unit_is_iso {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the unit is an isomorphism, then the left adjoint is full

Equations
theorem category_theory.L_faithful_of_unit_is_iso {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the unit is an isomorphism, then the left adjoint is faithful

noncomputable def category_theory.R_full_of_counit_is_iso {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the counit is an isomorphism, then the right adjoint is full

Equations
theorem category_theory.R_faithful_of_counit_is_iso {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :

If the counit is an isomorphism, then the right adjoint is faithful

@[protected, instance]
def category_theory.whisker_left_counit_iso_of_L_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
@[protected, instance]
def category_theory.whisker_right_counit_iso_of_L_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
@[protected, instance]
def category_theory.whisker_left_unit_iso_of_R_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
@[protected, instance]
def category_theory.whisker_right_unit_iso_of_R_fully_faithful {C : Type u₁} {D : Type u₂} {L : C D} {R : D C} (h : L R)  :
def category_theory.adjunction.restrict_fully_faithful {C : Type u₁} {D : Type u₂} {C' : Type u₃} {D' : Type u₄} (iC : C C') (iD : D D') {L' : C' D'} {R' : D' C'} (adj : L' R') {L : C D} {R : D C} (comm1 : iC L' L iD) (comm2 : iD R' R iC)  :
L R

If C is a full subcategory of C' and D is a full subcategory of D', then we can restrict an adjunction L' ⊣ R' where L' : C' ⥤ D' and R' : D' ⥤ C' to C and D. The construction here is slightly more general, in that C is required only to have a full and faithful "inclusion" functor iC : C ⥤ C' (and similarly iD : D ⥤ D') which commute (up to natural isomorphism) with the proposed restrictions.

Equations