-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense
Goal:
C₀ : Type u₀
C : Type u
inst : Category.{v₀, u₀} C₀
inst_1 : Category.{v, u} C
F : Functor.{v₀, v, u₀, u} C₀ C
J₀ : GrothendieckTopology.{v₀, u₀} C₀
J : GrothendieckTopology.{v, u} C
A : Type u'
inst_2 : Category.{v', u'} A
inst_3 : IsDenseSubsite J₀ J F
data : (X : C) → OneHypercoverDenseData.{u_1, v₀, v, u₀, u} F J₀ J X
G : Functor.{v, v', u, u'} Cᵒᵖ A
hG₀ : Presheaf.IsSheaf J₀ (F.op ⋙ G)
hG : (X : C) → IsLimit ((data X).toOneHypercover.multifork G)
X : C
S : J.Cover X
s : Multifork (S.index G)
i : (data X).I₀
W₀ : C₀
a : W₀ ⟶ (data X).toPreOneHypercoverDenseData.2 i
ha :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)) ↦
            Membership.mem (γ := Set (Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)))
              ((J₀ : (a : C₀) → Set (Sieve a)) ((data X).toPreOneHypercoverDenseData.2 i)) S)
            (Sieve.functorPullback F (Sieve.pullback ((data X).f i) (↑S : Sieve X))) ⋯) :
        Sieve ((data X).toPreOneHypercoverDenseData.2 i)).arrows
    a
Z₀ : C₀
b : Z₀ ⟶ (data X).toPreOneHypercoverDenseData.2 i
hb :
  (↑(Subtype.mk (p := fun (S : Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)) ↦
            Membership.mem (γ := Set (Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)))
              ((J₀ : (a : C₀) → Set (Sieve a)) ((data X).toPreOneHypercoverDenseData.2 i)) S)
            (Sieve.functorPullback F (Sieve.pullback ((data X).f i) (↑S : Sieve X))) ⋯) :
        Sieve ((data X).toPreOneHypercoverDenseData.2 i)).arrows
    b
U₀ : C₀
p₁ :
  U₀ ⟶
    { Y := W₀, f := a, hf := ha :
        GrothendieckTopology.Cover.Arrow
          (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)) ↦
            Membership.mem (γ := Set (Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)))
              ((J₀ : (a : C₀) → Set (Sieve a)) ((data X).toPreOneHypercoverDenseData.2 i)) S)
            (Sieve.functorPullback F (Sieve.pullback ((data X).f i) (↑S : Sieve X))) ⋯) }.Y
p₂ :
  U₀ ⟶
    { Y := Z₀, f := b, hf := hb :
        GrothendieckTopology.Cover.Arrow
          (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)) ↦
            Membership.mem (γ := Set (Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)))
              ((J₀ : (a : C₀) → Set (Sieve a)) ((data X).toPreOneHypercoverDenseData.2 i)) S)
            (Sieve.functorPullback F (Sieve.pullback ((data X).f i) (↑S : Sieve X))) ⋯) }.Y
fac :
  p₁ ≫
      { Y := W₀, f := a, hf := ha :
          GrothendieckTopology.Cover.Arrow
            (Subtype.mk (p := fun (S : Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)) ↦
              Membership.mem (γ := Set (Sieve.{v₀, u₀} (C := C₀) ((data X).toPreOneHypercoverDenseData.2 i)))
                ((J₀ : (a : C₀) → Set (Sieve a)) ((data X).toPreOneHypercoverDenseData.2 i)) S)
              (Sieve.functorPullback F (Sieve.pullback ((data X).f i) (↑S : Sieve X))) ⋯) }.f =
    p₂ ≫ { Y := Z₀, f := b, hf := hb }.f
⊢ F.map p₁ ≫ { Y := F.obj W₀, f := F.map a ≫ (data X).f i, hf := ha : S.Arrow }.f =
    F.map p₂ ≫ { Y := F.obj Z₀, f := F.map b ≫ (data X).f i, hf := hb }.f

Tactic:
simp only [← Functor.map_comp_assoc, fac]

Premises used: CategoryTheory.Functor.map_comp_assoc