-- 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 s
inst_8 : SmallCategory K
inst_9 : FinCategory K
inst_10 : HasLimitsOfShape.{s, s, t, w} 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 : C
S : Cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))
k : K
rw :
  CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
      (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) S)
      (CategoryStruct.comp (Y := (J.plusObj (limit F)).obj (op X))
        (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (Iso.hom
              (HasLimit.isoOfNatIso
                (Iso.symm
                  (NatIso.ofComponents
                    (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯))) :
            limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) ⟶
              limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
              (Iso.inv (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
                limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
              (Iso.inv
                  (HasColimit.isoOfNatIso
                    (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
                      (limit.isLimit (F ⋙ J.diagramFunctor D X)))) :
                colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (limit F)).obj (op X)) :
            limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (limit F)).obj (op X)))
        (NatTrans.app (J.plusMap (limit.π F k)) (op X))) =
    IsLimit.lift (limit.isLimit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))) S ≫
      (limit.cone (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X))).π.app k
⊢ CategoryStruct.comp (Y := (J.plusObj (limit F)).obj (op X))
      (CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
        (limit.lift (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) S)
        (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (Iso.hom
              (HasLimit.isoOfNatIso
                (Iso.symm
                  (NatIso.ofComponents
                    (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯))) :
            limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) ⟶
              limit (colimit (F ⋙ J.diagramFunctor D X).flip))
          (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
              (Iso.inv (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
                limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
              (Iso.inv
                  (HasColimit.isoOfNatIso
                    (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
                      (limit.isLimit (F ⋙ J.diagramFunctor D X)))) :
                colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (limit F)).obj (op X)) :
            limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (limit F)).obj (op X))))
      (NatTrans.app (J.plusMap (limit.π F k)) (op X)) =
    S.π.app k

Tactic:
rw [← (limit.isLimit (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))).fac S k,
    Category.assoc]

Premises used: CategoryTheory.Category.assoc, Opposite, CategoryTheory.Functor.comp, CategoryTheory.GrothendieckTopology.plusFunctor, CategoryTheory.Limits.limit.isLimit, Opposite.op, CategoryTheory.evaluation, CategoryTheory.Functor.obj, CategoryTheory.Limits.IsLimit.fac