-- Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction
Goal:
X : Truncated 2
C D : Type u
inst : SmallCategory C
inst_1 : SmallCategory D
F : X.HomotopyCategory ⥤ C
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))
h' :
  ∀ {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))
x₀ x₁ x₂ : X.obj (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋0⦌ Edge._proof_1))
e₀₁ : Edge x₀ x₁
e₁₂ : Edge x₁ x₂
e₀₂ : Edge x₀ x₂
h : e₀₁.CompStruct e₁₂ e₀₂
h₂ :
  ComposableArrows.mk₁ (F.map (homMk e₀₁)) =
    (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 2 _proof_2 _proof_5).op) :
        (a : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_5 })) →
          ((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := _proof_2 }))
      y
h₀ :
  ComposableArrows.mk₁ (F.map (homMk e₁₂)) =
    (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 0 _proof_2 _proof_5).op) :
        (a : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_5 })) →
          ((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := _proof_2 }))
      y
h₁ :
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C))
            (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 0) ⋯ _proof_4))) :
        (a : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_4 })) →
          ((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := ⋯ }))
      (ComposableArrows.mk₂ (F.map (homMk e₀₁)) (F.map (homMk e₁₂))) =
    ComposableArrows.mk₁ (F.map (homMk e₁₂))
⊢ 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⦌ ⋯)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C))
              (Quiver.Hom.op (Hom.tr (SimplexCategory.mkOfSucc 1) ⋯ _proof_4))) :
          (a : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_4 })) →
            ((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := ⋯ }))
        (ComposableArrows.mk₂ (F.map (homMk e₀₁)) (F.map (homMk e₁₂))) :
      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⦌ ⋯)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C))
              (Quiver.Hom.op (Hom.tr (SimplexCategory.mkOfSucc 1) ⋯ _proof_4))) :
          (a : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_4 })) →
            ((truncation 2).obj (nerve C)).obj (op { obj := ⦋1⦌, property := ⋯ }))
        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) ⦋1⦌ ⋯)))

Tactic:
simp only [SimplexCategory.mkOfSucc_one_eq_δ, ← h₀]

Premises used: SimplexCategory.mkOfSucc_one_eq_δ