-- 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
j : (J.Cover (unop (op X)))ᵒᵖ
e_a :
  CategoryStruct.comp (Y := colimit (J.diagram (limit F) X)) (colimit.ι (J.diagram (limit F) X) j)
      (colimMap (J.diagramNatTrans (limit.π F k) X)) =
    CategoryStruct.comp (Y := colimit (J.diagram (limit F) X)) (colimit.ι (J.diagram (limit F) X) j)
      (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
        (Iso.hom
            (HasColimit.isoOfNatIso
              (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
                (limit.isLimit (F ⋙ J.diagramFunctor D X)))) :
          (J.plusObj (limit F)).obj (op X) ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
        (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
            (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
              colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ limit (colimit (F ⋙ J.diagramFunctor D X).flip))
            (CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
                (Iso.inv
                  (HasLimit.isoOfNatIso
                    (Iso.symm
                      (NatIso.ofComponents
                        (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯))))
                (limit.π (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) k) :
              limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (F.obj k)).obj (op X)) :
          colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (F.obj k)).obj (op X)))
⊢ Eq (α := Functor.obj (J.diagram (limit F) (unop (op X))) j ⟶ (J.plusObj (F.obj k)).obj (op X))
    (colimit.ι (J.diagram (limit F) (unop (op X))) j ≫ NatTrans.app (J.plusMap (limit.π F k)) (op X))
    (CategoryStruct.comp (Y := colimit (J := (J.Cover (unop (op X)))ᵒᵖ) (J.diagram (limit F) (unop (op X))))
      (colimit.ι (J.diagram (limit F) (unop (op X))) j)
      (CategoryStruct.comp (Y := colimit (limit (F ⋙ J.diagramFunctor D X)))
        (Iso.hom
            (HasColimit.isoOfNatIso
              (IsLimit.conePointUniqueUpToIso (isLimitOfPreserves (J.diagramFunctor D X) (limit.isLimit F))
                (limit.isLimit (F ⋙ J.diagramFunctor D X)))) :
          (J.plusObj (limit F)).obj (op X) ⟶ colimit (limit (F ⋙ J.diagramFunctor D X)))
        (CategoryStruct.comp (Y := limit (colimit (F ⋙ J.diagramFunctor D X).flip))
            (Iso.hom (colimitLimitIso (F ⋙ J.diagramFunctor D X)) :
              colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ limit (colimit (F ⋙ J.diagramFunctor D X).flip))
            (CategoryStruct.comp (Y := limit (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)))
                (Iso.inv
                  (HasLimit.isoOfNatIso
                    (Iso.symm
                      (NatIso.ofComponents
                        (fun (k : K) ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D X).flip k) ⋯))))
                (limit.π (F ⋙ J.plusFunctor D ⋙ Functor.obj (evaluation Cᵒᵖ D) (op X)) k) :
              limit (colimit (F ⋙ J.diagramFunctor D X).flip) ⟶ (J.plusObj (F.obj k)).obj (op X)) :
          colimit (limit (F ⋙ J.diagramFunctor D X)) ⟶ (J.plusObj (F.obj k)).obj (op X))))

Tactic:
dsimp [plusMap]

Premises used: CategoryTheory.GrothendieckTopology.plusMap