-- 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
h :
  Eq (α := (↑(Functor.obj (Discrete.functor fun (j : J) ↦ Y.obj { as := j }) { as := j : Discrete J }) : 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)))
        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 := limit Y) (Y :=
            Functor.obj (Discrete.functor fun (j : J) ↦ Y.obj { as := j }) { as := j : Discrete J })
            (CategoryStruct.comp (Y := limit (J := Discrete J) (Discrete.functor fun (j : J) ↦ Y.obj { as := j }))
              (IsLimit.lift (productIsProduct' Y) (getLimitCone Y).cone)
              (limit.π (Discrete.functor fun (j : J) ↦ Y.obj { as := j }) { as := j })) :
          (a : (↑(limit Y) : Type u)) →
            (↑((Discrete.functor fun (j : J) ↦ Y.obj { as := j }).obj { as := j }) : Type u))
        (((AddCommGrpCat.Hom.hom' (limit.lift Y ((Cone.postcompose f).obj (Pi.cone X))) :
                (↑(∏ᶜ fun (j : J) ↦ X.obj { as := j }) : Type u) →+ (↑(limit Y) : Type u)) :
              (a : (↑(∏ᶜ fun (j : J) ↦ X.obj { as := j }) : Type u)) → (↑(limit Y) : Type u))
            (((AddCommGrpCat.Hom.hom'
                      (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) }) :
                    (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u) →+
                      (↑(∏ᶜ fun (j : J) ↦ X.obj { as := j }) : Type u)) :
                  (a : (↑(AddCommGrpCat.of ((i : J) → (↑(X.obj { as := i }) : Type u))) : Type u)) →
                    (↑(∏ᶜ fun (j : J) ↦ X.obj { as := j }) : Type u))
                g :
              (↑(∏ᶜ fun (j : J) ↦ X.obj { as := j }) : Type u)) :
          (↑(limit Y) : Type u)) :
      (↑(Functor.obj (Discrete.functor fun (j : J) ↦ Y.obj { as := j }) { as := j : Discrete J }) : Type u))
⊢ Eq (α := (↑(Y.obj { as := j }) : 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)))
        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 :=
              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))
        g j :
      (↑(Y.obj { as := j }) : Type u))

Tactic:
change _ = (_ ≫ limit.π (Discrete.functor fun j ↦ Y.obj { as := j }) ⟨j⟩) _

Premises used: CategoryTheory.Limits.limit.π, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Discrete.functor, CategoryTheory.Discrete.mk, CategoryTheory.CategoryStruct.comp, DFunLike.coe, CategoryTheory.Functor.obj