-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.Plus (goal)
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
⊢ 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))
have 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) sorry :
        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)) sorry)
          (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) sorry)
    (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 := sorry })
        sorry)
      (Multiequalizer.ι (Cover.index (Cover.pullback ⊤ f.unop) P) I))
Tactic: dsimp [Cover.Arrow.base]