-- Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u_1
J : Type u_2
inst_1 : Category.{v_1, u_1} D
inst_2 : Category.{v_2, u_2} J
inst_3 : HasLimitsOfShape.{v_2, u_2, v, u} J C
inst_4 : HasLimitsOfShape.{v_2, u_2, v_1, u_1} J D
inst_5 : HasExactLimitsOfShape.{v_1, v_2, u_1, u_2} J D
F : Functor.{v, v_1, u, u_1} C D
inst_6 : PreservesFiniteColimits F
inst_7 : ReflectsFiniteColimits F
inst_8 : HasFiniteColimits.{v, u} C
inst_9 : PreservesLimitsOfShape.{v_2, u_2, v, v_1, u, u_1} J F
I : Type
inst_10 : SmallCategory I
inst_11 : FinCategory I
G : I ⥤ Functor.{v_2, v, u_2, u} J C
c : Cocone G
hc : IsColimit c
i : I
val :
  Eq (α := F.obj (limit (G.obj i)) ⟶ F.obj (limit c.pt))
    (CategoryStruct.comp (Y := limit (G.obj i ⋙ F))
      (Iso.hom (preservesLimitIso F (G.obj i)) : F.obj (limit (G.obj i)) ⟶ limit (G.obj i ⋙ F))
      (CategoryStruct.comp (Y := limit (c.pt ⋙ F))
          (limMap (whiskerRight (c.ι.app i) F) : limit (G.obj i ⋙ F) ⟶ limit (c.pt ⋙ F))
          (Iso.inv (preservesLimitIso F c.pt) : limit (c.pt ⋙ F) ⟶ F.obj (limit c.pt)) :
        limit (G.obj i ⋙ F) ⟶ F.obj (limit c.pt)))
    (F.map (limMap (c.ι.app i)))
⊢ Eq (α := (G ⋙ lim ⋙ F).obj i ⟶ Cocone.pt (F := G ⋙ lim ⋙ F) (F.mapCocone (mapCocone lim c) : Cocone ((G ⋙ lim) ⋙ F)))
    (CategoryStruct.comp (Y :=
      Cocone.pt (F := G ⋙ lim ⋙ F)
        (obj (C := Cocone (G ⋙ obj (whiskeringRight J C D) F ⋙ lim))
          (Cocone.precompose (Iso.inv (G.isoWhiskerLeft (Iso.symm (preservesLimitNatIso F)))))
          (mapCocone (obj (whiskeringRight J C D) F ⋙ lim) c)))
      (NatTrans.app
          (Cocone.ι
            (obj (C := Cocone (G ⋙ obj (whiskeringRight J C D) F ⋙ lim))
              (Cocone.precompose (Iso.inv (G.isoWhiskerLeft (Iso.symm (preservesLimitNatIso F)))))
              (mapCocone (obj (whiskeringRight J C D) F ⋙ lim) c)))
          i :
        (G ⋙ lim ⋙ F).obj i ⟶
          obj.{0, v_1, 0, u_1}
            (obj (const I)
              (Cocone.pt (F := G ⋙ lim ⋙ F)
                (obj (C := Cocone (G ⋙ obj (whiskeringRight J C D) F ⋙ lim))
                  (Cocone.precompose (Iso.inv (G.isoWhiskerLeft (Iso.symm (preservesLimitNatIso F)))))
                  (mapCocone (obj (whiskeringRight J C D) F ⋙ lim) c))))
            i)
      (Iso.hom (Iso.app (Iso.symm (preservesLimitNatIso F)) c.pt)))
    (NatTrans.app (Cocone.ι (F.mapCocone (mapCocone lim c))) i)

Tactic:
simp only [Functor.comp_obj, lim_obj, Functor.mapCocone_pt, isoWhiskerLeft_inv, Iso.symm_inv,
        Cocone.precompose_obj_pt, whiskeringRight_obj_obj, Functor.const_obj_obj,
        Cocone.precompose_obj_ι, NatTrans.comp_app, whiskerLeft_app, preservesLimitNatIso_hom_app,
        Functor.mapCocone_ι_app, Functor.comp_map, whiskeringRight_obj_map, lim_map, Iso.app_hom,
        Iso.symm_hom, preservesLimitNatIso_inv_app, Category.assoc]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.Functor.mapCocone_ι_app, CategoryTheory.Limits.lim_map, CategoryTheory.Iso.symm_hom, CategoryTheory.Functor.whiskerLeft_app, CategoryTheory.Iso.app_hom, CategoryTheory.preservesLimitNatIso_inv_app, CategoryTheory.Iso.symm_inv, CategoryTheory.Limits.lim_obj, CategoryTheory.Functor.isoWhiskerLeft_inv, CategoryTheory.Limits.Cocone.precompose_obj_ι, CategoryTheory.Functor.whiskeringRight_obj_map, CategoryTheory.Functor.whiskeringRight_obj_obj, CategoryTheory.Functor.const_obj_obj, CategoryTheory.NatTrans.comp_app, CategoryTheory.preservesLimitNatIso_hom_app, CategoryTheory.Functor.mapCocone_pt, CategoryTheory.Functor.comp_map, CategoryTheory.Limits.Cocone.precompose_obj_pt