[GOAL]
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝ : C
⊢ { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
          map := fun {X Y} f =>
            NatTrans.mk fun X_1 =>
              { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                map_add' :=
                  (_ :
                    ∀
                      (g g' :
                        ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj X_1)),
                      (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
      (𝟙 x✝) =
    𝟙
      ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun X_1 =>
                { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                              X_1)),
                        (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
        x✝)
[PROOFSTEP]
ext
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝² : C
x✝¹ : Cᵒᵖ
x✝ :
  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun X_1 =>
                  { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                X_1)),
                          (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
          x✝²).obj
      x✝¹)
⊢ ↑(NatTrans.app
          ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
                map := fun {X Y} f =>
                  NatTrans.mk fun X_1 =>
                    { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                      map_add' :=
                        (_ :
                          ∀
                            (g g' :
                              ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                  X_1)),
                            (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
            (𝟙 x✝²))
          x✝¹)
      x✝ =
    ↑(NatTrans.app
          (𝟙
            ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun X_1 =>
                      { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat)
                                        X).obj
                                    X_1)),
                              (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
              x✝²))
          x✝¹)
      x✝
[PROOFSTEP]
dsimp
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝² : C
x✝¹ : Cᵒᵖ
x✝ :
  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun X_1 =>
                  { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                X_1)),
                          (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
          x✝²).obj
      x✝¹)
⊢ x✝ ≫ 𝟙 x✝² = x✝
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
          map := fun {X Y} f =>
            NatTrans.mk fun X_1 =>
              { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                map_add' :=
                  (_ :
                    ∀
                      (g g' :
                        ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj X_1)),
                      (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
      (f ≫ g) =
    { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun X_1 =>
                { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                              X_1)),
                        (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
        f ≫
      { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun X_1 =>
                { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                              X_1)),
                        (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
        g
[PROOFSTEP]
ext
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
x✝¹ : Cᵒᵖ
x✝ :
  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun X_1 =>
                  { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                X_1)),
                          (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
          X✝).obj
      x✝¹)
⊢ ↑(NatTrans.app
          ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
                map := fun {X Y} f =>
                  NatTrans.mk fun X_1 =>
                    { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                      map_add' :=
                        (_ :
                          ∀
                            (g g' :
                              ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                  X_1)),
                            (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
            (f ≫ g))
          x✝¹)
      x✝ =
    ↑(NatTrans.app
          ({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun X_1 =>
                      { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat)
                                        X).obj
                                    X_1)),
                              (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
              f ≫
            { obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun X_1 =>
                      { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat)
                                        X).obj
                                    X_1)),
                              (g + g') ≫ f = g ≫ f + g' ≫ f) } }.map
              g)
          x✝¹)
      x✝
[PROOFSTEP]
dsimp
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
x✝¹ : Cᵒᵖ
x✝ :
  ↑(({ obj := fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun X_1 =>
                  { toZeroHom := { toFun := fun g => g ≫ f, map_zero' := (_ : 0 ≫ f = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun Y => preadditiveYonedaObj Y ⋙ forget₂ (ModuleCat (End Y)) AddCommGroupCat) X).obj
                                X_1)),
                          (g + g') ≫ f = g ≫ f + g' ≫ f) } }.obj
          X✝).obj
      x✝¹)
⊢ x✝ ≫ f ≫ g = (x✝ ≫ f) ≫ g
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝ : Cᵒᵖ
⊢ { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
          map := fun {X Y} f =>
            NatTrans.mk fun Y_1 =>
              { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                map_add' :=
                  (_ :
                    ∀
                      (g g' :
                        ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                            Y_1)),
                      f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
      (𝟙 x✝) =
    𝟙
      ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun Y_1 =>
                { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                              Y_1)),
                        f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
        x✝)
[PROOFSTEP]
ext
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝² : Cᵒᵖ
x✝¹ : C
x✝ :
  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun Y_1 =>
                  { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                                Y_1)),
                          f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
          x✝²).obj
      x✝¹)
⊢ ↑(NatTrans.app
          ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
                map := fun {X Y} f =>
                  NatTrans.mk fun Y_1 =>
                    { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                      map_add' :=
                        (_ :
                          ∀
                            (g g' :
                              ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat)
                                      X).obj
                                  Y_1)),
                            f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
            (𝟙 x✝²))
          x✝¹)
      x✝ =
    ↑(NatTrans.app
          (𝟙
            ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun Y_1 =>
                      { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat)
                                        X).obj
                                    Y_1)),
                              f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
              x✝²))
          x✝¹)
      x✝
[PROOFSTEP]
dsimp
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
x✝² : Cᵒᵖ
x✝¹ : C
x✝ :
  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun Y_1 =>
                  { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                                Y_1)),
                          f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
          x✝²).obj
      x✝¹)
⊢ 𝟙 x✝².unop ≫ x✝ = x✝
[PROOFSTEP]
simp
[GOAL]
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : Cᵒᵖ
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
⊢ { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
          map := fun {X Y} f =>
            NatTrans.mk fun Y_1 =>
              { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                map_add' :=
                  (_ :
                    ∀
                      (g g' :
                        ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                            Y_1)),
                      f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
      (f ≫ g) =
    { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun Y_1 =>
                { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                              Y_1)),
                        f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
        f ≫
      { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
            map := fun {X Y} f =>
              NatTrans.mk fun Y_1 =>
                { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                  map_add' :=
                    (_ :
                      ∀
                        (g g' :
                          ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                              Y_1)),
                        f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
        g
[PROOFSTEP]
ext
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : Cᵒᵖ
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
x✝¹ : C
x✝ :
  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun Y_1 =>
                  { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                                Y_1)),
                          f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
          X✝).obj
      x✝¹)
⊢ ↑(NatTrans.app
          ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
                map := fun {X Y} f =>
                  NatTrans.mk fun Y_1 =>
                    { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                      map_add' :=
                        (_ :
                          ∀
                            (g g' :
                              ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat)
                                      X).obj
                                  Y_1)),
                            f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
            (f ≫ g))
          x✝¹)
      x✝ =
    ↑(NatTrans.app
          ({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun Y_1 =>
                      { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat)
                                        X).obj
                                    Y_1)),
                              f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
              f ≫
            { obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
                  map := fun {X Y} f =>
                    NatTrans.mk fun Y_1 =>
                      { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                        map_add' :=
                          (_ :
                            ∀
                              (g g' :
                                ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat)
                                        X).obj
                                    Y_1)),
                              f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.map
              g)
          x✝¹)
      x✝
[PROOFSTEP]
dsimp
[GOAL]
case w.h.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
X✝ Y✝ Z✝ : Cᵒᵖ
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
x✝¹ : C
x✝ :
  ↑(({ obj := fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat,
              map := fun {X Y} f =>
                NatTrans.mk fun Y_1 =>
                  { toZeroHom := { toFun := fun g => f.unop ≫ g, map_zero' := (_ : f.unop ≫ 0 = 0) },
                    map_add' :=
                      (_ :
                        ∀
                          (g g' :
                            ↑(((fun X => preadditiveCoyonedaObj X ⋙ forget₂ (ModuleCat (End X)) AddCommGroupCat) X).obj
                                Y_1)),
                          f.unop ≫ (g + g') = f.unop ≫ g + f.unop ≫ g') } }.obj
          X✝).obj
      x✝¹)
⊢ (g.unop ≫ f.unop) ≫ x✝ = g.unop ≫ f.unop ≫ x✝
[PROOFSTEP]
simp
