-- 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 :
  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 })))
          h.simplex) :
      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 })))
          h.simplex) :
      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))))
dsimp :
  ComposableArrows.mk₁ (X₀ :=
      F.obj
        (mk
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op) :
              (a : X.obj (op { obj := ⦋1⦌, property := Edge._proof_3 })) →
                X.obj (op { obj := ⦋0⦌, property := Edge._proof_1 }))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (δ₂ 1 _proof_2 _proof_5).op) :
                  (a : X.obj (op { obj := ⦋2⦌, property := _proof_5 })) →
                    X.obj (op { obj := ⦋1⦌, property := _proof_2 }))
                h.simplex :
              X.obj
                (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋1⦌ _proof_2))))))
      (F.map
        (homMk
          (Edge.mk'
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (δ₂ 1 _proof_2 _proof_5).op) :
                (a : X.obj (op { obj := ⦋2⦌, property := _proof_5 })) → X.obj (op { obj := ⦋1⦌, property := _proof_2 }))
              h.simplex)))) =
    (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 1 _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
⊢ 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 1 _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 })))
          h.simplex) :
      ComposableArrows C 1)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (CategoryTheory.Functor.map (Functor.obj (truncation 2) (nerve C)) (δ₂ 1 _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))))

Tactic:
dsimp at h₀ h₂ ⊢