-- 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
dsimp :
  Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop Z))ᵒᵖ) (J.diagram P (unop Z)))
    (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))
        ⋯)
      (colimit.ι (J.diagram P (unop Z)) (op ((unop S).pullback (g.unop ≫ f.unop)))))
    (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)) ⋯)
      (CategoryStruct.comp (Y :=
        Functor.obj
          (Functor.comp (D := (J.Cover (unop Z))ᵒᵖ)
            { obj := fun (X : (J.Cover (unop Y))ᵒᵖ) ↦ op ((unop X).pullback g.unop),
              map := fun {X Y_1 : (J.Cover (unop Y))ᵒᵖ} (f : X ⟶ Y_1) ↦ ((J.pullback g.unop).map f.unop).op,
              map_id := ⋯, map_comp := ⋯ }
            (J.diagram P (unop Z)))
          (op ((unop S).pullback f.unop)))
        (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))
          ⋯)
        (colimit.ι (J.diagram P (unop Z)) (op (((unop S).pullback f.unop).pullback g.unop)))))
⊢ Eq (α := multiequalizer ((unop S).index P) ⟶ colimit (J := (J.Cover (unop Z))ᵒᵖ) (J.diagram P (unop Z)))
    (CategoryStruct.comp (Y := ((J.pullback (g.unop ≫ f.unop)).op ⋙ J.diagram P (unop Z)).obj S)
      (Multiequalizer.lift ((unop ((J.pullback (g.unop ≫ f.unop)).op.obj S)).index P)
        (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : (unop ((J.pullback (g.unop ≫ f.unop)).op.obj S)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (colimit.ι (J.diagram P (unop Z)) ((J.pullback (g.unop ≫ f.unop)).op.obj S)))
    (CategoryStruct.comp (Y := ((J.pullback f.unop).op ⋙ J.diagram P (unop Y)).obj S)
      (Multiequalizer.lift ((unop ((J.pullback f.unop).op.obj S)).index P) (Functor.obj (J.diagram P (unop X)) S)
        (fun (I : (unop ((J.pullback f.unop).op.obj S)).shape.L) ↦
          Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I))
        ⋯)
      (CategoryStruct.comp (Y := ((J.pullback g.unop).op ⋙ J.diagram P (unop Z)).obj ((J.pullback f.unop).op.obj S))
        (Multiequalizer.lift ((unop ((J.pullback g.unop).op.obj ((J.pullback f.unop).op.obj S))).index P)
          (Functor.obj (J.diagram P (unop Y)) ((J.pullback f.unop).op.obj S))
          (fun (I : (unop ((J.pullback g.unop).op.obj ((J.pullback f.unop).op.obj S))).shape.L) ↦
            Multiequalizer.ι ((unop ((J.pullback f.unop).op.obj S)).index P) (Cover.Arrow.base I))
          ⋯)
        (colimit.ι (J.diagram P (unop Z)) ((J.pullback g.unop).op.obj ((J.pullback f.unop).op.obj S)))))

Tactic:
dsimp only [Functor.op, pullback_obj]

Premises used: CategoryTheory.Functor.op, CategoryTheory.GrothendieckTopology.pullback_obj