-- 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 })))
g : (i : J) → (↑(X.obj { as := i }) : Type u)
j : J
⊢ Eq (α := (↑(Y.obj { as := j }) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat.{u}) ↦ (↑x1 : Type u) →+ (↑x2 : Type u)) (CC :=
            AddCommGrpCat.carrier) (X := X.obj { as := j }) (f.app { as := j }) :
          (a : (↑(X.obj { as := j }) : Type u)) → (↑(Y.obj { as := j }) : Type u))
        (g j) :
      (↑(Y.obj { as := j }) : 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 :=
              Y.obj { as := j })
              (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)))
                      (limit.π Y { as := j }) :
                    (∏ᶜ fun (j : J) ↦ X.obj { as := j }) ⟶ Y.obj { as := j : Discrete J }) :
                AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i : Discrete J }) : Type u)) ⟶ Y.obj { as := j }) :
            (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u) →+
              (↑(Y.obj { as := j }) : Type u)) :
          (a : (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u)) →
            (↑(Y.obj { as := j }) : Type u))
        g :
      (↑(Y.obj { as := j }) : Type u))

Tactic:
simp