-- Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction
Goal:
X : Truncated 2
C D : Type u
inst : SmallCategory C
inst_1 : SmallCategory D
F : X.HomotopyCategory ⥤ C
x : X.obj (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋2⦌ _proof_4))
y :
  Functor.obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.obj (truncation 2) (nerve C))
    (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋2⦌ _proof_4))
h₂ :
  Eq (α :=
    Functor.obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.obj (truncation 2) (nerve C))
      (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2)))
    ((fun
            (f :
              X.obj
                (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2))) ↦
          ComposableArrows.mk₁ (F.map (homMk (Edge.mk' f))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (δ₂ 2 _proof_2 _proof_5).op) :
            (a : (fun (X : Type u) ↦ X) (X.obj (op { obj := ⦋1 + 1⦌, property := _proof_5 }))) →
              (fun (X : Type u) ↦ X) (X.obj (op { obj := ⦋1⦌, property := _proof_2 })))
          x) :
      ComposableArrows C 1)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 2 _proof_2 _proof_5).op) :
          (a :
              (fun (X : Type u) ↦ X)
                (((truncation 2).obj (nerve C)).obj (op { obj := ⦋1 + 1⦌, property := _proof_5 }))) →
            (fun (X : Type u) ↦ X) (((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := _proof_2 })))
        y :
      (fun (X : Type u) ↦ X)
        (Functor.obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.obj (truncation 2) (nerve C))
          (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2))))
h₀ :
  Eq (α :=
    Functor.obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.obj (truncation 2) (nerve C))
      (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2)))
    ((fun
            (f :
              X.obj
                (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2))) ↦
          ComposableArrows.mk₁ (F.map (homMk (Edge.mk' f))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (δ₂ 0 _proof_2 _proof_5).op) :
            (a : (fun (X : Type u) ↦ X) (X.obj (op { obj := ⦋1 + 1⦌, property := _proof_5 }))) →
              (fun (X : Type u) ↦ X) (X.obj (op { obj := ⦋1⦌, property := _proof_2 })))
          x) :
      ComposableArrows C 1)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 0 _proof_2 _proof_5).op) :
          (a :
              (fun (X : Type u) ↦ X)
                (((truncation 2).obj (nerve C)).obj (op { obj := ⦋1 + 1⦌, property := _proof_5 }))) →
            (fun (X : Type u) ↦ X) (((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := _proof_2 })))
        y :
      (fun (X : Type u) ↦ X)
        (Functor.obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.obj (truncation 2) (nerve C))
          (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2))))
⊢ ∀ {a b : X.obj (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ ⋯))}
    (e : Edge a b), ComposableArrows.mk₁ (F.map (homMk (Edge.mk' e.edge))) = ComposableArrows.mk₁ (F.map (homMk e))

Tactic:
have h' {a b : X _⦋0⦌₂} (e : Edge a b) :
          ComposableArrows.mk₁ (F.map (homMk (Edge.mk' e.edge))) =
            ComposableArrows.mk₁ (F.map (homMk e)) :=
        ComposableArrows.arrowEquiv.injective
          (congr_arg F.mapArrow.obj (congr_arrowMk_homMk (Edge.mk' e.edge) e rfl))