-- Module: Mathlib.CategoryTheory.Sites.LeftExact
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
D : Type w
inst_1 : Category.{t, w} D
inst_2 : ∀ (P : Functor.{v, t, u, w} Cᵒᵖ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst_3 : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst_4 : (X Y : D) → FunLike.{u_1 + 1, t + 1, t + 1} (FD X Y) (CD X) (CD Y)
inst_5 : ConcreteCategory D FD (CC := CD)
inst_6 : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D (FC := FD) (CC := CD))
inst_7 : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape K D
inst_11 : PreservesLimitsOfShape K (forget D (FC := FD) (CC := CD))
inst_12 : ReflectsLimitsOfShape K (forget D (FC := FD) (CC := CD))
F : K ⥤ Functor.{v, t, u, w} Cᵒᵖ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
m :
  S.pt ⟶
    Cone.pt (F := F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X)
      (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))
hm :
  ∀ (j : K),
    m ≫ NatTrans.app (Cone.π (Functor.mapCone (J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) X) (limit.cone F))) j =
      S.π.app j
k_1 : K
k : (J.Cover (unop X))ᵒᵖ
preservesLimit :
  Eq (α :=
    multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
    (CategoryStruct.comp (Y := Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k)
        (NatTrans.app
            (Iso.hom
              (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
                (limit.isLimit (F ⋙ J.diagramFunctor D (unop X)))))
            k :
          Functor.obj (J.diagram (limit F) (unop X)) k ⟶ Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k)
        (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
            (colimit.ι (limit (F ⋙ J.diagramFunctor D (unop X))) k :
              Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k ⟶
                colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
            (CategoryStruct.comp (Y :=
                limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
                (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))) :
                  colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
                    limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
                (limit.π (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) k_1 ≫
                    Iso.hom (colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ J.diagramFunctor D (unop X))) k_1) :
                  limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) ⟶
                    (J.plusObj (F.obj k_1)).obj X) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
                colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X))) :
          Functor.obj (limit (F ⋙ J.diagramFunctor D (unop X))) k ⟶
            colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X))) :
      multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
    (NatTrans.app (J.diagramNatTrans (limit.π F k_1) (unop X)) k ≫ colimit.ι (J.diagram (F.obj k_1) (unop X)) k :
      Functor.obj (J.diagram (limit F) (unop X)) k ⟶
        colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
⊢ Eq (α :=
    multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
        (NatTrans.app
              (Iso.hom
                (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F))
                  (limit.isLimit (F ⋙ J.diagramFunctor D (unop X)))))
              k ≫
            colimit.ι (limit (F ⋙ J.diagramFunctor D (unop X))) k :
          multiequalizer ((unop k).index (limit F)) ⟶
            colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))))
        (CategoryStruct.comp (Y :=
            limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
            (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))) :
              colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
                limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))))
            (limit.π (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) k_1 ≫
                Iso.hom (colimitObjIsoColimitCompEvaluation (Functor.flip (F ⋙ J.diagramFunctor D (unop X))) k_1) :
              limit (colimit (J := (J.Cover (unop X))ᵒᵖ) (Functor.flip (F ⋙ J.diagramFunctor D (unop X)))) ⟶
                (J.plusObj (F.obj k_1)).obj X) :
          colimit (J := (J.Cover (unop X))ᵒᵖ) (limit (F ⋙ J.diagramFunctor D (unop X))) ⟶
            colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X))) :
      multiequalizer ((unop k).index (limit F)) ⟶ colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))
    (NatTrans.app (J.diagramNatTrans (limit.π F k_1) (unop X)) k ≫ colimit.ι (J.diagram (F.obj k_1) (unop X)) k :
      Functor.obj (J.diagram (limit F) (unop X)) k ⟶
        colimit (J := (J.Cover (unop X))ᵒᵖ) (J.diagram (F.obj k_1) (unop X)))

Tactic:
simp only [Category.assoc]

Premises used: CategoryTheory.Category.assoc