-- 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
x : Arrow (Sheaf.{v, v', u, u'} J A)
rw :
  Presheaf.IsLocallySurjective J (FA := FA) (CA := CA) (G :=
    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 x.hom.hom))
    (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 x.hom.hom))
⊢ IsLocallySurjective (FA := FA) (CA := CA) (F₂ :=
    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 x.hom.hom)))
    (NatTrans.app (inv (Adjunction.counit (sheafificationAdjunction J A))) x.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 x.hom.hom)))

Tactic:
rw [← isLocallySurjective_sheafToPresheaf_map_iff, Functor.map_comp,
      Presheaf.comp_isLocallySurjective_iff, isLocallySurjective_sheafToPresheaf_map_iff,
      Presheaf.isLocallySurjective_presheafToSheaf_map_iff]

Premises used: CategoryTheory.Sheaf.isLocallySurjective_sheafToPresheaf_map_iff, CategoryTheory.Presheaf.comp_isLocallySurjective_iff, CategoryTheory.Functor.map_comp, CategoryTheory.Presheaf.isLocallySurjective_presheafToSheaf_map_iff