[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.24757, u_1} J
inst✝¹ : Category.{?u.24761, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
⊢ whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.24757, u_1} J
inst✝¹ : Category.{?u.24761, u_2} C
inst✝ : HasZeroMorphisms C
X✝ Y✝ : J ⥤ ShortComplex C
φ : X✝ ⟶ Y✝
⊢ whiskerRight φ π₁ ≫
      ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) (_ : whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0))
          Y✝).f =
    ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) (_ : whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0))
          X✝).f ≫
      whiskerRight φ π₂
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.24757, u_1} J
inst✝¹ : Category.{?u.24761, u_2} C
inst✝ : HasZeroMorphisms C
X✝ Y✝ : J ⥤ ShortComplex C
φ : X✝ ⟶ Y✝
⊢ whiskerRight φ π₂ ≫
      ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) (_ : whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0))
          Y✝).g =
    ((fun F => mk (whiskerLeft F π₁Toπ₂) (whiskerLeft F π₂Toπ₃) (_ : whiskerLeft F π₁Toπ₂ ≫ whiskerLeft F π₂Toπ₃ = 0))
          X✝).g ≫
      whiskerRight φ π₃
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : ShortComplex (J ⥤ C)
⊢ ∀ {X Y : J} (f : X ⟶ Y),
    ((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₁.map f ≫
        ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₁.obj x)) Y).hom =
      ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₁.obj x)) X).hom ≫
        ((functor J C ⋙ inverse J C).obj x✝).X₁.map f
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : ShortComplex (J ⥤ C)
⊢ ∀ {X Y : J} (f : X ⟶ Y),
    ((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₂.map f ≫
        ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₂.obj x)) Y).hom =
      ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₂.obj x)) X).hom ≫
        ((functor J C ⋙ inverse J C).obj x✝).X₂.map f
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : ShortComplex (J ⥤ C)
⊢ ∀ {X Y : J} (f : X ⟶ Y),
    ((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₃.map f ≫
        ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₃.obj x)) Y).hom =
      ((fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₃.obj x)) X).hom ≫
        ((functor J C ⋙ inverse J C).obj x✝).X₃.map f
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : ShortComplex (J ⥤ C)
⊢ (NatIso.ofComponents fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₁.obj x)).hom ≫
      ((functor J C ⋙ inverse J C).obj x✝).f =
    ((𝟭 (ShortComplex (J ⥤ C))).obj x✝).f ≫
      (NatIso.ofComponents fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₂.obj x)).hom
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : ShortComplex (J ⥤ C)
⊢ (NatIso.ofComponents fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₂.obj x)).hom ≫
      ((functor J C ⋙ inverse J C).obj x✝).g =
    ((𝟭 (ShortComplex (J ⥤ C))).obj x✝).g ≫
      (NatIso.ofComponents fun x => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x✝).X₃.obj x)).hom
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.53514, u_1} J
inst✝¹ : Category.{?u.53518, u_2} C
inst✝ : HasZeroMorphisms C
⊢ ∀ {X Y : ShortComplex (J ⥤ C)} (f : X ⟶ Y),
    (𝟭 (ShortComplex (J ⥤ C))).map f ≫
        ((fun x =>
              isoMk (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₁.obj x_1))
                (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₂.obj x_1))
                (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₃.obj x_1)))
            Y).hom =
      ((fun x =>
              isoMk (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₁.obj x_1))
                (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₂.obj x_1))
                (NatIso.ofComponents fun x_1 => Iso.refl (((𝟭 (ShortComplex (J ⥤ C))).obj x).X₃.obj x_1)))
            X).hom ≫
        (functor J C ⋙ inverse J C).map f
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.111338, u_1} J
inst✝¹ : Category.{?u.111342, u_2} C
inst✝ : HasZeroMorphisms C
x✝¹ : J ⥤ ShortComplex C
x✝ : J
⊢ (Iso.refl (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).X₁).hom ≫ (((𝟭 (J ⥤ ShortComplex C)).obj x✝¹).obj x✝).f =
    (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).f ≫ (Iso.refl (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).X₂).hom
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.111338, u_1} J
inst✝¹ : Category.{?u.111342, u_2} C
inst✝ : HasZeroMorphisms C
x✝¹ : J ⥤ ShortComplex C
x✝ : J
⊢ (Iso.refl (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).X₂).hom ≫ (((𝟭 (J ⥤ ShortComplex C)).obj x✝¹).obj x✝).g =
    (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).g ≫ (Iso.refl (((inverse J C ⋙ functor J C).obj x✝¹).obj x✝).X₃).hom
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.111338, u_1} J
inst✝¹ : Category.{?u.111342, u_2} C
inst✝ : HasZeroMorphisms C
x✝ : J ⥤ ShortComplex C
⊢ ∀ {X Y : J} (f : X ⟶ Y),
    ((inverse J C ⋙ functor J C).obj x✝).map f ≫
        ((fun x =>
              isoMk (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₁)
                (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₂)
                (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₃))
            Y).hom =
      ((fun x =>
              isoMk (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₁)
                (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₂)
                (Iso.refl (((inverse J C ⋙ functor J C).obj x✝).obj x).X₃))
            X).hom ≫
        ((𝟭 (J ⥤ ShortComplex C)).obj x✝).map f
[PROOFSTEP]
aesop_cat
[GOAL]
J : Type u_1
C : Type u_2
inst✝² : Category.{?u.111338, u_1} J
inst✝¹ : Category.{?u.111342, u_2} C
inst✝ : HasZeroMorphisms C
⊢ ∀ {X Y : J ⥤ ShortComplex C} (f : X ⟶ Y),
    (inverse J C ⋙ functor J C).map f ≫
        ((fun x =>
              NatIso.ofComponents fun x_1 =>
                isoMk (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₁)
                  (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₂)
                  (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₃))
            Y).hom =
      ((fun x =>
              NatIso.ofComponents fun x_1 =>
                isoMk (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₁)
                  (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₂)
                  (Iso.refl (((inverse J C ⋙ functor J C).obj x).obj x_1).X₃))
            X).hom ≫
        (𝟭 (J ⥤ ShortComplex C)).map f
[PROOFSTEP]
aesop_cat
