-- Module: Mathlib.CategoryTheory.Sites.DenseSubsite.Basic
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
D : Type u_2
inst_1 : Category.{v_2, u_2} D
E : Type u_3
inst_2 : Category.{v_3, u_3} E
J : GrothendieckTopology.{v_1, u_1} C
K : GrothendieckTopology.{v_2, u_2} D
L : GrothendieckTopology.{v_3, u_3} E
G : Functor.{v_1, v_2, u_1, u_2} C D
A : Type u_4
inst_3 : Category.{v_4, u_4} A
inst_4 : IsDenseSubsite J K G
F : Sheaf.{v_1, v_4, u_1, u_4} J A
X Y : C
f : G.obj X ⟶ G.obj Y
W₀ : C
a : W₀ ⟶ X
ha :
  (↑(Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯) :
        Sieve X).arrows
    a
W₀' : C
a' : W₀' ⟶ X
ha' :
  (↑(Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯) :
        Sieve X).arrows
    a'
T₀ : C
p₁ :
  T₀ ⟶
    { Y := W₀, f := a, hf := ha :
        GrothendieckTopology.Cover.Arrow
          (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯) }.Y
p₂ :
  T₀ ⟶
    { Y := W₀', f := a', hf := ha' :
        GrothendieckTopology.Cover.Arrow
          (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f) ⋯) }.Y
fac :
  p₁ ≫
      { Y := W₀, f := a, hf := ha :
          GrothendieckTopology.Cover.Arrow
            (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
              ⋯) }.f =
    p₂ ≫ { Y := W₀', f := a', hf := ha' }.f
h :
  G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₁ ≫
      G.map (Exists.choose (p := fun (l : W₀ ⟶ Y) ↦ G.map l = G.map a ≫ f) ha) =
    G.map
      ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
        Exists.choose (p := fun (l : W₀' ⟶ Y) ↦ G.map l = G.map a' ≫ f) ha')
h₁ :
  G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₁ ≫
      G.map (Exists.choose (p := fun (l : W₀ ⟶ Y) ↦ G.map l = G.map a ≫ f) ha) =
    G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
      G.map (Exists.choose (p := fun (l : W₀' ⟶ Y) ↦ G.map l = G.map a' ≫ f) ha')
h₂ :
  G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₁ ≫
      G.map a ≫ f =
    G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
      G.map (Exists.choose (p := fun (l : W₀' ⟶ Y) ↦ G.map l = G.map a' ≫ f) ha')
h₃ :
  G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₁ ≫
      G.map a ≫ f =
    G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
      G.map a' ≫ f
h₄ :
  G.map
        ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
              { Y := W₀, f := a, hf := ha :
                    GrothendieckTopology.Cover.Arrow
                      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                        (G.imageSieve f) ⋯) }.Relation
                { Y := W₀', f := a', hf := ha' :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) } }.g₁ ≫
          a) ≫
      f =
    G.map
        { Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
      G.map a' ≫ f
h₅ :
  G.map
        ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
              { Y := W₀, f := a, hf := ha :
                    GrothendieckTopology.Cover.Arrow
                      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                        (G.imageSieve f) ⋯) }.Relation
                { Y := W₀', f := a', hf := ha' :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) } }.g₁ ≫
          a) ≫
      f =
    G.map
        ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
              { Y := W₀, f := a, hf := ha :
                    GrothendieckTopology.Cover.Arrow
                      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                        (G.imageSieve f) ⋯) }.Relation
                { Y := W₀', f := a', hf := ha' :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) } }.g₂ ≫
          a') ≫
      f
h₆ h₇ :
  G.map
        (p₂ ≫
          { Y := W₀', f := a', hf := ha' :
              GrothendieckTopology.Cover.Arrow
                (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                  ⋯) }.f) ≫
      f =
    G.map
        ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
              { Y := W₀, f := a, hf := ha :
                    GrothendieckTopology.Cover.Arrow
                      (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                        (G.imageSieve f) ⋯) }.Relation
                { Y := W₀', f := a', hf := ha' :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) } }.g₂ ≫
          a') ≫
      f
⊢ G.map
      ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₁ ≫
        Exists.choose (p := fun (l : W₀ ⟶ Y) ↦ G.map l = G.map a ≫ f) ha) =
    G.map
      ({ Z := T₀, g₁ := p₁, g₂ := p₂, w := fac :
            { Y := W₀, f := a, hf := ha :
                  GrothendieckTopology.Cover.Arrow
                    (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X)
                      (G.imageSieve f) ⋯) }.Relation
              { Y := W₀', f := a', hf := ha' :
                GrothendieckTopology.Cover.Arrow
                  (Subtype.mk (p := fun (S : Sieve.{v_1, u_1} X) ↦ S ∈ (J : (a : C) → Set (Sieve a)) X) (G.imageSieve f)
                    ⋯) } }.g₂ ≫
        Exists.choose (p := fun (l : W₀' ⟶ Y) ↦ G.map l = G.map a' ≫ f) ha')

Tactic:
rw [Functor.map_comp, Functor.map_comp, ha.choose_spec, ha'.choose_spec,
        ← Functor.map_comp_assoc, ← Functor.map_comp_assoc, fac]

Premises used: Exists.choose_spec, CategoryTheory.Functor.map_comp_assoc, CategoryTheory.Functor.map_comp