-- Module: Mathlib.CategoryTheory.Adjunction.Additive
Goal:
C : Type u₁
D : Type u₂
inst : Category.{v₁, u₁} C
inst_1 : Category.{v₂, u₂} D
inst_2 : Preadditive.{v₁, u₁} C
inst_3 : Preadditive.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
G : Functor.{v₂, v₁, u₂, u₁} D C
adj : F ⊣ G
inst_4 : F.Additive
Y : D
X Y_1 : Cᵒᵖ
g : X ⟶ Y_1
y : (↑(obj (obj preadditiveYoneda (G.obj Y)) X) : Type v₁)
⊢ Eq (α :=
    (↑(((preadditiveYoneda ⋙
                  obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                    obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat) AddCommGrpCat.uliftFunctor).obj
              Y).obj
          Y_1) :
      Type (max v₁ v₂)))
    ((AddCommGrpCat.Hom.hom
            (((G ⋙
                        preadditiveYoneda ⋙
                          obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat) AddCommGrpCat.uliftFunctor).obj
                    Y).map
                g ≫
              Iso.hom
                ((fun (X : Cᵒᵖ) ↦
                    (AddEquiv.toAddCommGrpIso
                        (AddEquiv.trans AddEquiv.ulift
                          (AddEquiv.trans (AddEquiv.symm (adj.homAddEquiv (unop X) Y))
                            (AddEquiv.symm AddEquiv.ulift))) :
                      ((G ⋙
                                  preadditiveYoneda ⋙
                                    obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat)
                                      AddCommGrpCat.uliftFunctor).obj
                              Y).obj
                          X ≅
                        ((preadditiveYoneda ⋙
                                  obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                                    obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat)
                                      AddCommGrpCat.uliftFunctor).obj
                              Y).obj
                          X))
                  Y_1)) :
          (a :
              (↑(((G ⋙
                            preadditiveYoneda ⋙
                              (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj AddCommGrpCat.uliftFunctor).obj
                        Y).obj
                    X) :
                Type (max v₁ v₂))) →
            (↑(((preadditiveYoneda ⋙
                          (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat).obj F.op ⋙
                            (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj AddCommGrpCat.uliftFunctor).obj
                      Y).obj
                  Y_1) :
              Type (max v₁ v₂)))
        { down := y } :
      (↑(((preadditiveYoneda ⋙
                    obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                      obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat) AddCommGrpCat.uliftFunctor).obj
                Y).obj
            Y_1) :
        Type (max v₁ v₂)))
    ((AddCommGrpCat.Hom.hom
            (Iso.hom
                ((fun (X : Cᵒᵖ) ↦
                    (AddEquiv.toAddCommGrpIso
                        (AddEquiv.trans AddEquiv.ulift
                          (AddEquiv.trans (AddEquiv.symm (adj.homAddEquiv (unop X) Y))
                            (AddEquiv.symm AddEquiv.ulift))) :
                      ((G ⋙
                                  preadditiveYoneda ⋙
                                    obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat)
                                      AddCommGrpCat.uliftFunctor).obj
                              Y).obj
                          X ≅
                        ((preadditiveYoneda ⋙
                                  obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                                    obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat)
                                      AddCommGrpCat.uliftFunctor).obj
                              Y).obj
                          X))
                  X) ≫
              ((preadditiveYoneda ⋙
                        obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                          obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat) AddCommGrpCat.uliftFunctor).obj
                    Y).map
                g) :
          (a :
              (↑(((G ⋙
                            preadditiveYoneda ⋙
                              (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj AddCommGrpCat.uliftFunctor).obj
                        Y).obj
                    X) :
                Type (max v₁ v₂))) →
            (↑(((preadditiveYoneda ⋙
                          (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat).obj F.op ⋙
                            (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj AddCommGrpCat.uliftFunctor).obj
                      Y).obj
                  Y_1) :
              Type (max v₁ v₂)))
        { down := y } :
      (↑(((preadditiveYoneda ⋙
                    obj (whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat) F.op ⋙
                      obj (whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat) AddCommGrpCat.uliftFunctor).obj
                Y).obj
            Y_1) :
        Type (max v₁ v₂)))

Tactic:
exact AddEquiv.ulift.injective (adj.homEquiv_naturality_left_symm g.unop y)

Premises used: AddEquiv.ulift, Quiver.Hom.unop, AddEquiv.injective, CategoryTheory.Adjunction.homEquiv_naturality_left_symm