English
The forgetful functor forget X has a right adjoint (via costarAdjForget X) when binary coproducts exist.
Русский
Функтор забывания forget X имеет правый сопряжённый (через costarAdjForget X) при существовании бинарных coproducts.
LaTeX
$$forget X is RightAdjoint$$
Lean4
/-- The pushout along a mono that's preserved under pushouts is faithful.
This "preserved under pushouts" condition is automatically satisfied in abelian categories:
```
example [Abelian C] [Mono f] : (pushout f).Faithful := inferInstance
```
-/
instance faithful_pushout {X Y : C} (f : X ⟶ Y) [HasPushoutsAlong f] [∀ Z (g : X ⟶ Z), Mono (pushout.inl g f)] :
(pushout f).Faithful :=
by
have (Z : Under X) : Mono ((mapPushoutAdj f).unit.app Z) := by simp; infer_instance
exact (mapPushoutAdj f).faithful_L_of_mono_unit_app
