-- Module: Mathlib.CategoryTheory.Sites.Plus
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{w', w} D
inst_2 : ∀ (P : Functor.{v, w', u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Functor.{v, w', u, w} Cᵒᵖ D
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X Y Z : Cᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
S : (J.Cover (unop X))ᵒᵖ
e : (unop S).pullback (g.unop ≫ f.unop) ≅ ((unop S).pullback f.unop).pullback g.unop :=
  (unop S).pullbackComp g.unop f.unop
e_a :
  ∀ (I : (unop (op (((unop S).pullback f.unop).pullback g.unop))).shape.L),
    Eq (α :=
      Functor.obj (J.diagram P (unop X)) S ⟶ ((unop (op (((unop S).pullback f.unop).pullback g.unop))).index P).left I)
      (CategoryStruct.comp (Y := multiequalizer ((unop (op (((unop S).pullback f.unop).pullback g.unop))).index P))
        (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop Z))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (g.unop ≫ f.unop)),
                map := fun {X_1 Y_1 : (J.Cover (unop X))ᵒᵖ} (f_1 : X_1 ⟶ Y_1) ↦
                  ((J.pullback (g.unop ≫ f.unop)).map f_1.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop Z)))
            S)
          (Multiequalizer.lift (((unop S).pullback (g.unop ≫ f.unop)).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback (g.unop ≫ f.unop)).shape.L) ↦
              Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
            ⋯)
          (Functor.map (J.diagram P (unop Z)) e.inv.op))
        (Multiequalizer.ι ((unop (op (((unop S).pullback f.unop).pullback g.unop))).index P) I))
      (CategoryStruct.comp (Y := multiequalizer ((unop (op (((unop S).pullback f.unop).pullback g.unop))).index P))
        (CategoryStruct.comp (Y :=
          Functor.obj
            (Functor.comp (D := (J.Cover (unop Y))ᵒᵖ)
              { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback f.unop),
                map := fun {X_1 Y_1 : (J.Cover (unop X))ᵒᵖ} (f_1 : X_1 ⟶ Y_1) ↦ ((J.pullback f.unop).map f_1.unop).op,
                map_id := ⋯, map_comp := ⋯ }
              (J.diagram P (unop Y)))
            S)
          (Multiequalizer.lift (((unop S).pullback f.unop).index P) (Functor.obj (J.diagram P (unop X)) S)
            (fun (I : ((unop S).pullback f.unop).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I)) ⋯)
          (Multiequalizer.lift ((((unop S).pullback f.unop).pullback g.unop).index P)
            (Functor.obj (J.diagram P (unop Y)) (op ((unop S).pullback f.unop)))
            (fun (I : (((unop S).pullback f.unop).pullback g.unop).shape.L) ↦
              Multiequalizer.ι (((unop S).pullback f.unop).index P) (Cover.Arrow.base I))
            ⋯))
        (Multiequalizer.ι ((unop (op (((unop S).pullback f.unop).pullback g.unop))).index P) I))
⊢ Eq (α :=
    Functor.obj (J.diagram P (unop X)) S ⟶
      Functor.obj (J.diagram P (unop Z)) (op (((unop S).pullback f.unop).pullback g.unop)))
    (CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop Z))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback (g.unop ≫ f.unop)),
            map := fun {X_1 Y_1 : (J.Cover (unop X))ᵒᵖ} (f_1 : X_1 ⟶ Y_1) ↦
              ((J.pullback (g.unop ≫ f.unop)).map f_1.unop).op,
            map_id := ⋯, map_comp := ⋯ }
          (J.diagram P (unop Z)))
        S)
      (Multiequalizer.lift (((unop S).pullback (g.unop ≫ f.unop)).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback (g.unop ≫ f.unop)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (Functor.map (J.diagram P (unop Z)) e.inv.op))
    (CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.comp (D := (J.Cover (unop Y))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((unop X_1).pullback f.unop),
            map := fun {X_1 Y_1 : (J.Cover (unop X))ᵒᵖ} (f_1 : X_1 ⟶ Y_1) ↦ ((J.pullback f.unop).map f_1.unop).op,
            map_id := ⋯, map_comp := ⋯ }
          (J.diagram P (unop Y)))
        S)
      (Multiequalizer.lift (((unop S).pullback f.unop).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : ((unop S).pullback f.unop).shape.L) ↦ Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I)) ⋯)
      (Multiequalizer.lift ((((unop S).pullback f.unop).pullback g.unop).index P)
        (Functor.obj (J.diagram P (unop Y)) (op ((unop S).pullback f.unop)))
        (fun (I : (((unop S).pullback f.unop).pullback g.unop).shape.L) ↦
          Multiequalizer.ι (((unop S).pullback f.unop).index P) (Cover.Arrow.base I))
        ⋯))

Tactic:
refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)

Premises used: CategoryTheory.Limits.Multiequalizer.hom_ext