-- 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)
dsimp :
  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)))
⊢ locallySurjective J A (FA := FA) (CA := CA) (X := obj Arrow.leftFunc x) (Y :=
    (mapArrow (sheafToPresheaf J A) ⋙
          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) ⋙
            presheafToSheaf J A).obj
      x)
    (NatTrans.app
        (whiskerLeft Arrow.leftFunc (inv (Adjunction.counit (sheafificationAdjunction J A))) ≫
          whiskerLeft (mapArrow (sheafToPresheaf J A))
            (whiskerRight
              (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))
              (presheafToSheaf J A)))
        x :
      obj Arrow.leftFunc x ⟶
        (mapArrow (sheafToPresheaf J A) ⋙
              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) ⋙
                presheafToSheaf J A).obj
          x)

Tactic:
dsimp [locallySurjective]

Premises used: CategoryTheory.Sheaf.locallySurjective