-- Module: Mathlib.CategoryTheory.Limits.Preserves.Grothendieck
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
H : Type u₂
inst_1 : Category.{v₂, u₂} H
J : Type u₃
inst_2 : Category.{v₃, u₃} J
F : C ⥤ Cat
K : J ⥤ Grothendieck F ⥤ H
inst_3 : ∀ (c : C), HasColimitsOfShape.{v₄, u₄, v₂, u₂} (↑(F.obj c) : Type u₄) H
inst_4 : HasLimitsOfShape.{v₃, u₃, v₂, u₂} J H
inst_5 : ∀ (c : C), PreservesLimitsOfShape (C := Functor.{v₄, v₂, u₄, u₂} (↑(F.obj c) : Type u₄) H) J colim
c₁ c₂ : C
f : c₁ ⟶ c₂
w :
  ∀ (j : (↑(F.obj c₁) : Type u₄)),
    Eq (α := (Grothendieck.ι F c₁ ⋙ limit K).obj j ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
      (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₁ ⋙ limit K)) (colimit.ι (Grothendieck.ι F c₁ ⋙ limit K) j)
          (CategoryStruct.comp (Y := colimit ((F.map f).toFunctor ⋙ Grothendieck.ι F c₂ ⋙ limit K))
            (colimMap
              (whiskerRight (Grothendieck.ιNatTrans f) (limit K) ≫
                ((F.map f).toFunctor.associator (Grothendieck.ι F c₂) (limit K)).hom))
            (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₂ ⋙ limit K))
              (colimit.pre (Grothendieck.ι F c₂ ⋙ limit K) (F.map f).toFunctor)
              (CategoryStruct.comp (Y :=
                colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
                (Iso.hom
                    (HasColimit.isoOfNatIso (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₂)))) :
                  colimit (Grothendieck.ι F c₂ ⋙ limit K) ⟶
                    colimit
                      (limit
                        (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
                (CategoryStruct.comp (Y :=
                    limit
                      ((K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙
                        colim))
                    (Iso.hom
                      (preservesLimitIso colim
                        (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
                    (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                      (Iso.hom
                          (HasLimit.isoOfNatIso
                            (K.associator
                                (obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))
                                colim ≪≫
                              K.isoWhiskerLeft (Iso.symm (fiberwiseColimCompEvaluationIso c₂)) ≪≫
                                (K.associator (fiberwiseColim F H) (obj (evaluation C H) c₂)).symm)) :
                        limit
                            ((K ⋙
                                obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙
                              colim) ⟶
                          limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                      (Iso.inv (limitObjIsoLimitCompEvaluation (K ⋙ fiberwiseColim F H) c₂) :
                        limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂) ⟶
                          (limit (K ⋙ fiberwiseColim F H)).obj c₂)) :
                  colimit
                      (limit
                        (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))) ⟶
                    (limit (K ⋙ fiberwiseColim F H)).obj c₂)))) :
        (Grothendieck.ι F c₁ ⋙ limit K).obj j ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
      (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₁ ⋙ limit K)) (colimit.ι (Grothendieck.ι F c₁ ⋙ limit K) j)
          (CategoryStruct.comp (Y :=
            colimit (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
            (Iso.hom (HasColimit.isoOfNatIso (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₁)))) :
              colimit (Grothendieck.ι F c₁ ⋙ limit K) ⟶
                colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
            (CategoryStruct.comp (Y :=
                limit
                  ((K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁)) ⋙ colim))
                (Iso.hom
                  (preservesLimitIso colim
                    (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
                (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₁))
                  (Iso.hom
                    (HasLimit.isoOfNatIso
                      (K.associator
                          (obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))
                          colim ≪≫
                        K.isoWhiskerLeft (Iso.symm (fiberwiseColimCompEvaluationIso c₁)) ≪≫
                          (K.associator (fiberwiseColim F H) (obj (evaluation C H) c₁)).symm)))
                  (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                    (limMap ((K ⋙ fiberwiseColim F H).whiskerLeft (map (evaluation C H) f)) :
                      limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₁) ⟶
                        limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                    (Iso.inv (limitObjIsoLimitCompEvaluation (K ⋙ fiberwiseColim F H) c₂) :
                      limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂) ⟶
                        (limit (K ⋙ fiberwiseColim F H)).obj c₂))) :
              colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))) ⟶
                (limit (K ⋙ fiberwiseColim F H)).obj c₂)) :
        (Grothendieck.ι F c₁ ⋙ limit K).obj j ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
⊢ Eq (α := colimit (Grothendieck.ι F c₁ ⋙ limit K) ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
    (CategoryStruct.comp (Y := colimit ((F.map f).toFunctor ⋙ Grothendieck.ι F c₂ ⋙ limit K))
        (colimMap
          (whiskerRight (Grothendieck.ιNatTrans f) (limit K) ≫
            ((F.map f).toFunctor.associator (Grothendieck.ι F c₂) (limit K)).hom))
        (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₂ ⋙ limit K))
          (colimit.pre (Grothendieck.ι F c₂ ⋙ limit K) (F.map f).toFunctor)
          (CategoryStruct.comp (Y :=
            colimit (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
            (Iso.hom (HasColimit.isoOfNatIso (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₂)))) :
              colimit (Grothendieck.ι F c₂ ⋙ limit K) ⟶
                colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
            (CategoryStruct.comp (Y :=
                limit
                  ((K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙ colim))
                (Iso.hom
                  (preservesLimitIso colim
                    (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
                (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                  (Iso.hom
                      (HasLimit.isoOfNatIso
                        (K.associator
                            (obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))
                            colim ≪≫
                          K.isoWhiskerLeft (Iso.symm (fiberwiseColimCompEvaluationIso c₂)) ≪≫
                            (K.associator (fiberwiseColim F H) (obj (evaluation C H) c₂)).symm)) :
                    limit
                        ((K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙
                          colim) ⟶
                      limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                  (Iso.inv (limitObjIsoLimitCompEvaluation (K ⋙ fiberwiseColim F H) c₂) :
                    limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂) ⟶
                      (limit (K ⋙ fiberwiseColim F H)).obj c₂)) :
              colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))) ⟶
                (limit (K ⋙ fiberwiseColim F H)).obj c₂))) :
      colimit (Grothendieck.ι F c₁ ⋙ limit K) ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
    (CategoryStruct.comp (Y :=
        colimit (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
        (Iso.hom (HasColimit.isoOfNatIso (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₁)))) :
          colimit (Grothendieck.ι F c₁ ⋙ limit K) ⟶
            colimit (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
        (CategoryStruct.comp (Y :=
            limit ((K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁)) ⋙ colim))
            (Iso.hom
              (preservesLimitIso colim
                (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
            (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₁))
              (Iso.hom
                (HasLimit.isoOfNatIso
                  (K.associator (obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))
                      colim ≪≫
                    K.isoWhiskerLeft (Iso.symm (fiberwiseColimCompEvaluationIso c₁)) ≪≫
                      (K.associator (fiberwiseColim F H) (obj (evaluation C H) c₁)).symm)))
              (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                (limMap ((K ⋙ fiberwiseColim F H).whiskerLeft (map (evaluation C H) f)) :
                  limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₁) ⟶
                    limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
                (Iso.inv (limitObjIsoLimitCompEvaluation (K ⋙ fiberwiseColim F H) c₂) :
                  limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂) ⟶
                    (limit (K ⋙ fiberwiseColim F H)).obj c₂))) :
          colimit (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))) ⟶
            (limit (K ⋙ fiberwiseColim F H)).obj c₂) :
      colimit (Grothendieck.ι F c₁ ⋙ limit K) ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)

Tactic:
apply colimit.hom_ext

Premises used: CategoryTheory.Limits.colimit.hom_ext