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

Tactic:
dsimp [Cover.Arrow.base]

Premises used: CategoryTheory.GrothendieckTopology.Cover.Arrow.base