-- Lean 4 proof step from Mathlib.CategoryTheory.Sites.CompatiblePlus (goal)
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type u_1
inst_1 : Category.{v_1, u_1} D
E : Type u_2
inst_2 : Category.{v_2, u_2} E
F : Functor.{v_1, v_2, u_1, u_2} D E
inst_3 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst_4 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst_5 : ∀ (X : C) (W : J.Cover X) (P : Functor.{v, v_1, u, u_1} Cᵒᵖ D), PreservesLimit (W.index P).multicospan F
inst_6 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst_7 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst_8 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Functor.{v, v_1, u, u_1} Cᵒᵖ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
    (CategoryStruct.comp (Y := F.obj (multiequalizer ((unop W).index Q)))
        (F.map
            (Multiequalizer.lift ((unop W).index Q) (multiequalizer ((unop W).index P))
              (fun (x : (unop W).Arrow) ↦ Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y)) ⋯) :
          F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (multiequalizer ((unop W).index Q)))
        (CategoryStruct.comp (Y := obj (J.diagram (Q ⋙ F) (unop X)) W)
            (NatTrans.app (Iso.hom (J.diagramCompIso F Q (unop X))) W :
              obj (J.diagram Q (unop X) ⋙ F) W ⟶ obj (J.diagram (Q ⋙ F) (unop X)) W)
            (colimit.ι (J.diagram (Q ⋙ F) (unop X)) W :
              obj (J.diagram (Q ⋙ F) (unop X)) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X))) :
          obj (J.diagram Q (unop X) ⋙ F) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
    (CategoryStruct.comp (Y := multiequalizer ((unop W).index (P ⋙ F)))
        (NatTrans.app (Iso.hom (J.diagramCompIso F P (unop X))) W :
          obj (J.diagram P (unop X) ⋙ F) W ⟶ obj (J.diagram (P ⋙ F) (unop X)) W)
        (CategoryStruct.comp (Y := multiequalizer ((unop W).index (Q ⋙ F)))
            (Multiequalizer.lift ((unop W).index (Q ⋙ F)) (multiequalizer ((unop W).index (P ⋙ F)))
              (fun (x : (unop W).Arrow) ↦ Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) ⋯)
            (colimit.ι (J.diagram (Q ⋙ F) (unop X)) W) :
          multiequalizer ((unop W).index (P ⋙ F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X))) :
      F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
have w : Eq (α := F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
  (CategoryStruct.comp (Y := obj (J.diagram (Q ⋙ F) (unop X)) W)
      (CategoryStruct.comp (Y := F.obj (multiequalizer ((unop W).index Q)))
          (F.map
              (Multiequalizer.lift ((unop W).index Q) (multiequalizer ((unop W).index P))
                (fun (x : (unop W).Arrow) ↦ Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y)) sorry) :
            F.obj (multiequalizer ((unop W).index P)) ⟶ F.obj (multiequalizer ((unop W).index Q)))
          (NatTrans.app (Iso.hom (J.diagramCompIso F Q (unop X))) W :
            obj (J.diagram Q (unop X) ⋙ F) W ⟶ obj (J.diagram (Q ⋙ F) (unop X)) W) :
        F.obj (multiequalizer ((unop W).index P)) ⟶ obj (J.diagram (Q ⋙ F) (unop X)) W)
      (colimit.ι (J.diagram (Q ⋙ F) (unop X)) W :
        obj (J.diagram (Q ⋙ F) (unop X)) W ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X))) :
    F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
  (CategoryStruct.comp (Y := multiequalizer ((unop W).index (Q ⋙ F)))
      (CategoryStruct.comp (Y := multiequalizer ((unop W).index (P ⋙ F)))
        (NatTrans.app (Iso.hom (J.diagramCompIso F P (unop X))) W :
          obj (J.diagram P (unop X) ⋙ F) W ⟶ obj (J.diagram (P ⋙ F) (unop X)) W)
        (Multiequalizer.lift ((unop W).index (Q ⋙ F)) (multiequalizer ((unop W).index (P ⋙ F)))
            (fun (x : (unop W).Arrow) ↦ Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) sorry :
          multiequalizer ((unop W).index (P ⋙ F)) ⟶ multiequalizer ((unop W).index (Q ⋙ F))))
      (colimit.ι (J.diagram (Q ⋙ F) (unop X)) W) :
    F.obj (multiequalizer ((unop W).index P)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (Q ⋙ F) (unop X)))
Tactic: simp only [← Category.assoc]