-- Module: Mathlib.CategoryTheory.Sites.EpiMono
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_1 : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst_2 : (X Y : A) → FunLike.{u_1 + 1, w + 1, w + 1} (FA X Y) (CA X) (CA Y)
inst_3 : ConcreteCategory.{w, v', u', u_1} A FA (CC := CA)
inst_4 : HasFunctorialSurjectiveInjectiveFactorization.{v', u', u_1, w} A (FC := FA) (CC := CA)
inst_5 : GrothendieckTopology.WEqualsLocallyBijective.{w, v', v, u', u, u_1} J A (FA := FA) (CA := CA)
data : FunctorialSurjectiveInjectiveFactorizationData.{v', u', u_1, w} A (FC := FA) (CC := CA)
inst_6 : HasWeakSheafify.{v, v', u, u'} J A
f : Arrow (Sheaf.{v, v', u, u'} J A)
w :
  Eq (α := obj (presheafToSheaf J A) (f.left.obj (P := Presheaf.IsSheaf J)) ⟶ f.right)
    (map (presheafToSheaf J A) (Arrow.mk f.hom.hom).hom ≫
      NatTrans.app (Adjunction.counit (sheafificationAdjunction J A)) f.right)
    (NatTrans.app (Adjunction.counit (sheafificationAdjunction J A)) f.left ≫ f.hom)
⊢ CategoryStruct.comp (Y :=
      obj (presheafToSheaf J A)
        (obj
          (MorphismProperty.FunctorialFactorizationData.Z (W₁ :=
            MorphismProperty.functorCategory (MorphismProperty.surjective A (FC := FA) (CC := CA)) Cᵒᵖ) (W₂ :=
            MorphismProperty.functorCategory (MorphismProperty.injective A (FC := FA) (CC := CA)) Cᵒᵖ)
            (MorphismProperty.FunctorialFactorizationData.functorCategory (W₁ :=
              MorphismProperty.surjective A (FC := FA) (CC := CA)) (W₂ :=
              MorphismProperty.injective A (FC := FA) (CC := CA)) data Cᵒᵖ))
          (Arrow.mk f.hom.hom)))
      (NatTrans.app (inv (Adjunction.counit (sheafificationAdjunction J A))) f.left ≫
        map (presheafToSheaf J A)
          (NatTrans.app
            (MorphismProperty.FunctorialFactorizationData.i (W₁ :=
              MorphismProperty.functorCategory (MorphismProperty.surjective A (FC := FA) (CC := CA)) Cᵒᵖ) (W₂ :=
              MorphismProperty.functorCategory (MorphismProperty.injective A (FC := FA) (CC := CA)) Cᵒᵖ)
              (MorphismProperty.FunctorialFactorizationData.functorCategory (W₁ :=
                MorphismProperty.surjective A (FC := FA) (CC := CA)) (W₂ :=
                MorphismProperty.injective A (FC := FA) (CC := CA)) data Cᵒᵖ))
            (Arrow.mk f.hom.hom)))
      (map (presheafToSheaf J A)
          (NatTrans.app
            (MorphismProperty.FunctorialFactorizationData.p (W₁ :=
              MorphismProperty.functorCategory (MorphismProperty.surjective A (FC := FA) (CC := CA)) Cᵒᵖ) (W₂ :=
              MorphismProperty.functorCategory (MorphismProperty.injective A (FC := FA) (CC := CA)) Cᵒᵖ)
              (MorphismProperty.FunctorialFactorizationData.functorCategory (W₁ :=
                MorphismProperty.surjective A (FC := FA) (CC := CA)) (W₂ :=
                MorphismProperty.injective A (FC := FA) (CC := CA)) data Cᵒᵖ))
            (Arrow.mk f.hom.hom)) ≫
        NatTrans.app (Adjunction.counit (sheafificationAdjunction J A)) f.right) =
    f.hom

Tactic:
simp only [assoc, ← Functor.map_comp_assoc,
      MorphismProperty.FunctorialFactorizationData.fac_app,
      NatIso.isIso_inv_app, IsIso.inv_comp_eq]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.IsIso.inv_comp_eq, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.MorphismProperty.FunctorialFactorizationData.fac_app, CategoryTheory.NatIso.isIso_inv_app