# mathlibdocumentation

@[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

@[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!)

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.

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