-- 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 : Cᵒᵖ
f : X ⟶ Y
e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := homOfLE ⋯
rw :
  Eq (α := P.obj X ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y)))
    (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (P.map f ≫
          Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
            (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (Functor.map (J.diagram P (unop Y)) e.op) ≫
      colimit.ι (J.diagram P (unop Y)) (op ((J.pullback f.unop).obj ⊤)))
    (CategoryStruct.comp (Y :=
      Functor.obj (C := (J.Cover (unop X))ᵒᵖ)
        (Functor.comp (D := (J.Cover (unop Y))ᵒᵖ)
          { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((J.pullback f.unop).obj (unop X_1)),
            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)))
        (op ⊤))
      (CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
          (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
          (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯))
      (colimit.ι (J.diagram P (unop Y)) (op ((J.pullback f.unop).obj ⊤))))
⊢ Eq (α := P.obj X ⟶ colimit (J := (J.Cover (unop Y))ᵒᵖ) (J.diagram P (unop Y)))
    (P.map f ≫
      CategoryStruct.comp (Y :=
        multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤) (Cover.index ⊤ P))
        (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop Y)))
          (fun (I : (Cover.shape.{v, u} (C := C) (X := unop Y) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
        (colimit.ι (J.diagram P (unop Y)) (op ⊤)))
    (CategoryStruct.comp (Y :=
      multiequalizer (J := Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤) (Cover.index ⊤ P))
      (Multiequalizer.lift (Cover.index ⊤ P) (P.obj (op (unop X)))
        (fun (I : (Cover.shape.{v, u} (C := C) (X := unop X) (J := J) ⊤).L) ↦ P.map I.f.op) ⋯)
      (CategoryStruct.comp (Y :=
        Functor.obj (C := (J.Cover (unop X))ᵒᵖ)
          (Functor.comp (D := (J.Cover (unop Y))ᵒᵖ)
            { obj := fun (X_1 : (J.Cover (unop X))ᵒᵖ) ↦ op ((J.pullback f.unop).obj (unop X_1)),
              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)))
          (op ⊤))
        (Multiequalizer.lift (((J.pullback f.unop).obj ⊤).index P)
          (Functor.obj (C := (J.Cover (unop X))ᵒᵖ) (J.diagram P (unop X)) (op ⊤))
          (fun (I : ((J.pullback f.unop).obj ⊤).shape.L) ↦ Multiequalizer.ι (Cover.index ⊤ P) (Cover.Arrow.base I)) ⋯)
        (colimit.ι (J.diagram P (unop Y)) (op ((J.pullback f.unop).obj ⊤)))))

Tactic:
rw [← colimit.w _ e.op, ← Category.assoc, ← Category.assoc, ← Category.assoc]

Premises used: CategoryTheory.Category.assoc, Quiver.Hom.op, CategoryTheory.Limits.colimit.w