Module: Mathlib.CategoryTheory.Sites.LeftExact

Goal state before:
pp: case e_a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹¹ : Category.{t, w} D
inst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁸ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁷ : ConcreteCategory D FD
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁵ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (J.Cover (unop X))ᵒᵖ
⊢ colimit.ι (J.diagram (limit F) (unop X)) k ≫
      (HasColimit.isoOfNatIso
            ((isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F)).conePointUniqueUpToIso
              (limit.isLimit (F ⋙ J.diagramFunctor D (unop X))))).hom ≫
        (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))).hom ≫
          (HasLimit.isoOfNatIso
                (NatIso.ofComponents
                    (fun k ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D (unop X)).flip k) ⋯).symm).inv ≫
            limit.π (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X) k✝ =
    colimit.ι (J.diagram (limit F) (unop X)) k ≫ colimMap (J.diagramNatTrans (limit.π F k✝) (unop X))

Tactic:
erw [colimit.ι_map, colimit.ι_desc_assoc, limit.lift_π]

Goal state after:
case e_a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝¹¹ : Category.{t, w} D
inst✝¹⁰ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝⁹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
FD : D → D → Type u_1
CD : D → Type t
inst✝⁸ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁷ : ConcreteCategory D FD
inst✝⁶ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)
inst✝⁵ : ∀ (X : C), Small.{t, max u v} (J.Cover X)ᵒᵖ
K : Type t
inst✝⁴ : SmallCategory K
inst✝³ : FinCategory K
inst✝² : HasLimitsOfShape K D
inst✝¹ : PreservesLimitsOfShape K (forget D)
inst✝ : ReflectsLimitsOfShape K (forget D)
F : K ⥤ Cᵒᵖ ⥤ D
X : Cᵒᵖ
S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X)
m : S.pt ⟶ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).pt
hm : ∀ (j : K), m ≫ ((J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj X).mapCone (limit.cone F)).π.app j = S.π.app j
k✝ : K
k : (J.Cover (unop X))ᵒᵖ
⊢ ((Cocone.precompose
                  ((isLimitOfPreserves (J.diagramFunctor D (unop X)) (limit.isLimit F)).conePointUniqueUpToIso
                      (limit.isLimit (F ⋙ J.diagramFunctor D (unop X)))).hom).obj
              (colimit.cocone (limit (F ⋙ J.diagramFunctor D (unop X))))).ι.app
        k ≫
      (colimitLimitIso (F ⋙ J.diagramFunctor D (unop X))).hom ≫
        ((Cone.postcompose
                    (NatIso.ofComponents
                          (fun k ↦ colimitObjIsoColimitCompEvaluation (F ⋙ J.diagramFunctor D (unop X)).flip k)
                          ⋯).symm.inv).obj
                (limit.cone (colimit (F ⋙ J.diagramFunctor D (unop X)).flip))).π.app
          k✝ =
    (J.diagramNatTrans (limit.π F k✝) (unop X)).app k ≫ colimit.ι (J.diagram (F.obj k✝) (unop X)) k