-- 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
d : (↑(F.obj c₁) : Type u₄)
w :
  Eq (α := (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
    (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
        (CategoryStruct.comp (Y :=
            limit ((K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙ colim))
            (CategoryStruct.comp (Y :=
              colimit
                (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
              (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₂ ⋙ limit K))
                  (CategoryStruct.comp (Y := colimit ((F.map f).toFunctor ⋙ Grothendieck.ι F c₂ ⋙ limit K))
                    (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₁ ⋙ limit K))
                      (colimit.ι (Grothendieck.ι F c₁ ⋙ limit K) d)
                      (colimMap
                        (whiskerRight (Grothendieck.ιNatTrans f) (limit K) ≫
                          ((F.map f).toFunctor.associator (Grothendieck.ι F c₂) (limit K)).hom)))
                    (colimit.pre (Grothendieck.ι F c₂ ⋙ limit K) (F.map f).toFunctor))
                  (Iso.hom
                    (HasColimit.isoOfNatIso (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₂))))) :
                (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶
                  colimit
                    (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))))
              (Iso.hom
                  (preservesLimitIso colim
                    (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))) :
                colimit
                    (limit
                      (K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂))) ⟶
                  limit
                    ((K ⋙ obj (whiskeringLeft (↑(F.obj c₂) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₂)) ⋙
                      colim)))
            (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))) :
          (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ 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) :
      (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
    (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₂))
        (CategoryStruct.comp (Y := limit ((K ⋙ fiberwiseColim F H) ⋙ obj (evaluation C H) c₁))
            (CategoryStruct.comp (Y :=
              limit
                ((K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁)) ⋙ colim))
              (CategoryStruct.comp (Y :=
                colimit
                  (limit (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
                (CategoryStruct.comp (Y := colimit (Grothendieck.ι F c₁ ⋙ limit K))
                    (colimit.ι (Grothendieck.ι F c₁ ⋙ limit K) d)
                    (Iso.hom
                      (HasColimit.isoOfNatIso
                        (Iso.symm (limitCompWhiskeringLeftIsoCompLimit K (Grothendieck.ι F c₁))))) :
                  (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶
                    colimit
                      (limit
                        (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))))
                (Iso.hom
                    (preservesLimitIso colim
                      (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))) :
                  colimit
                      (limit
                        (K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁))) ⟶
                    limit
                      ((K ⋙ obj (whiskeringLeft (↑(F.obj c₁) : Type u₄) (Grothendieck F) H) (Grothendieck.ι F c₁)) ⋙
                        colim)))
              (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))))
            (limMap ((K ⋙ fiberwiseColim F H).whiskerLeft (map (evaluation C H) f))) :
          (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ 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) :
      (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)
⊢ Eq (α := (Grothendieck.ι F c₁ ⋙ limit K).obj d ⟶ (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) d)
        (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 d ⟶ (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) d)
        (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 d ⟶ (limit (K ⋙ fiberwiseColim F H)).obj c₂)

Tactic:
simp only [← Category.assoc]

Premises used: CategoryTheory.Category.assoc