-- Module: Mathlib.Algebra.Category.Grp.AB
Goal:
J : Type u
inst : SmallCategory J
inst_1 : IsFiltered J
X Y : Discrete J ⥤ AddCommGrpCat
f : X ⟶ Y
hf : Epi f
iX : limit X ≅ AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u)) :=
  Iso.trans (Y := ∏ᶜ fun (j : J) ↦ X.obj { as := j }) (Iso.symm (Pi.isoLimit X))
    (IsLimit.conePointUniqueUpToIso (limit.isLimit (Discrete.functor fun (j : J) ↦ X.obj { as := j }))
      (LimitCone.isLimit (AddCommGrpCat.HasLimit.productLimitCone fun (j : J) ↦ X.obj { as := j })))
iY : limit Y ≅ AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u)) :=
  Iso.trans (Y := ∏ᶜ fun (j : J) ↦ Y.obj { as := j }) (Iso.symm (Pi.isoLimit Y))
    (IsLimit.conePointUniqueUpToIso (limit.isLimit (Discrete.functor fun (j : J) ↦ Y.obj { as := j }))
      (LimitCone.isLimit (AddCommGrpCat.HasLimit.productLimitCone fun (j : J) ↦ Y.obj { as := j })))
simp :
  Eq (α := ((i : J) → (↑(X.obj { as := i }) : Type u)) → (i : J) → (↑(Y.obj { as := i }) : Type u))
    (Pi.map (α := fun (i : J) ↦ (↑(X.obj { as := i }) : Type u)) (β := fun (i : J) ↦ (↑(Y.obj { as := i }) : Type u))
      fun (i : J) ↦
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier) (X := X.obj { as := i }) (f.app { as := i })) :
        (a : (↑(X.obj { as := i }) : Type u)) → (↑(Y.obj { as := i }) : Type u)))
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier) (X := AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) (Y :=
            AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u)))
            (CategoryStruct.comp (Y := ∏ᶜ fun (j : J) ↦ X.obj { as := j })
                (limit.lift (Discrete.functor fun (j : J) ↦ X.obj { as := j })
                    { pt := AddCommGrpCat.of ((j : J) → (↑(X.obj { as := j }) : Type u)),
                      π :=
                        Discrete.natTrans fun (j : Discrete J) ↦
                          AddCommGrpCat.ofHom
                            (Pi.evalAddMonoidHom (fun (j : J) ↦ (↑(X.obj { as := j }) : Type u)) j.as) } :
                  Cone.pt (J := Discrete J) (F := Discrete.functor fun (j : J) ↦ X.obj { as := j : Discrete J })
                      { pt := AddCommGrpCat.of ((j : J) → (↑(X.obj { as := j : Discrete J }) : Type u)),
                        π :=
                          Discrete.natTrans fun (j : Discrete J) ↦
                            AddCommGrpCat.ofHom
                              (Pi.evalAddMonoidHom (fun (j : J) ↦ (↑(X.obj { as := j : Discrete J }) : Type u)) j.as) :
                        Cone (J := Discrete J) (C := AddCommGrpCat)
                          (Discrete.functor fun (j : J) ↦ X.obj { as := j : Discrete J }) } ⟶
                    limit (J := Discrete J) (Discrete.functor fun (j : J) ↦ X.obj { as := j }))
                (CategoryStruct.comp (Y := limit Y) (limit.lift Y ((Cone.postcompose f).obj (Pi.cone X)))
                    (CategoryStruct.comp (Y := ∏ᶜ fun (j : J) ↦ Y.obj { as := j })
                      (IsLimit.lift (productIsProduct' Y) (getLimitCone Y).cone)
                      (AddCommGrpCat.ofHom
                        {
                          toFun :=
                            fun
                              (x :
                                (↑(LimitCone.cone (getLimitCone (Discrete.functor fun (j : J) ↦ Y.obj { as := j }))).1 :
                                  Type u))
                              (j : J) ↦
                            (@ConcreteCategory.hom _ AddCommGrpCat.instCategory
                                  (fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u))
                                  AddCommGrpCat.carrier _ _
                                  (Cone.pt (J := Discrete J) (F := Discrete.functor fun (j : J) ↦ Y.obj { as := j })
                                    (LimitCone.cone (getLimitCone (Discrete.functor fun (j : J) ↦ Y.obj { as := j }))))
                                  _
                                  (NatTrans.app
                                    (Cone.π
                                      (LimitCone.cone
                                        (getLimitCone (Discrete.functor fun (j : J) ↦ Y.obj { as := j }))))
                                    { as := j }) :
                                (a :
                                    (↑(getLimitCone (Discrete.functor fun (j : J) ↦ Y.obj { as := j })).cone.pt :
                                      Type u)) →
                                  (↑(Y.obj { as := j }) : Type u))
                              x,
                          map_zero' := ⋯, map_add' := ⋯ })) :
                  (∏ᶜ fun (j : J) ↦ X.obj { as := j }) ⟶
                    AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i : Discrete J }) : Type u))) :
              AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i : Discrete J }) : Type u)) ⟶
                AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u))) :
          (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u) →+
            (↑(AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u))) : Type u)) :
      (a : (i : J) → (↑(X.obj { as := i }) : Type u)) → (i : J) → (↑(Y.obj { as := i }) : Type u))
⊢ Eq (α := ((i : J) → (↑(X.obj { as := i }) : Type u)) → (i : J) → (↑(Y.obj { as := i }) : Type u))
    (Pi.map (α := fun (i : J) ↦ (↑(X.obj { as := i }) : Type u)) (β := fun (i : J) ↦ (↑(Y.obj { as := i }) : Type u))
      fun (i : J) ↦
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier) (X := X.obj { as := i }) (f.app { as := i })) :
        (a : (↑(X.obj { as := i }) : Type u)) → (↑(Y.obj { as := i }) : Type u)))
    (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
          AddCommGrpCat.carrier) (Y := AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u)))
          (iX.inv ≫ CategoryTheory.Functor.map lim f ≫ iY.hom)) :
      (a : (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u)) →
        (↑(AddCommGrpCat.of ((i : J) → (↑(Y.obj { as := i }) : Type u))) : Type u))

Tactic:
simp only [Discrete.functor_obj_eq_as, Discrete.mk_as, Pi.isoLimit,
          IsLimit.conePointUniqueUpToIso, limit.cone, AddCommGrpCat.HasLimit.productLimitCone,
          Iso.trans_inv, Functor.mapIso_inv, IsLimit.uniqueUpToIso_inv, Cone.forget_map,
          IsLimit.liftConeMorphism_hom, limit.isLimit_lift, Iso.symm_inv, Functor.mapIso_hom,
          IsLimit.uniqueUpToIso_hom, lim_obj, lim_map, Iso.trans_hom, Iso.symm_hom,
          AddCommGrpCat.HasLimit.lift, Functor.const_obj_obj, Category.assoc, limit.lift_map_assoc,
          Pi.cone_pt, iX, iY]

Premises used: CategoryTheory.Discrete.mk_as, CategoryTheory.Category.assoc, AddCommGrpCat.HasLimit.productLimitCone, CategoryTheory.Limits.lim_map, CategoryTheory.Limits.IsLimit.uniqueUpToIso_inv, CategoryTheory.Limits.IsLimit.liftConeMorphism_hom, CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso, CategoryTheory.Limits.limit.isLimit_lift, CategoryTheory.Iso.symm_hom, CategoryTheory.Limits.limit.cone, CategoryTheory.Limits.IsLimit.uniqueUpToIso_hom, CategoryTheory.Iso.symm_inv, CategoryTheory.Discrete.functor_obj_eq_as, CategoryTheory.Limits.lim_obj, CategoryTheory.Limits.Pi.isoLimit, CategoryTheory.Iso.trans_hom, AddCommGrpCat.HasLimit.lift, CategoryTheory.Limits.Pi.cone_pt, CategoryTheory.Functor.mapIso_inv, CategoryTheory.Functor.const_obj_obj, CategoryTheory.Limits.Cone.forget_map, CategoryTheory.Functor.mapIso_hom, CategoryTheory.Iso.trans_inv, CategoryTheory.Limits.limit.lift_map_assoc