-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.Injective.Resolution (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : EnoughInjectives.{v, u} C
Z : C
⊢ ShortComplex.Exact.{v, u}
      (ShortComplex.mk (X₁ :=
        ShortComplex.X₂.{v, u}
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
            (Functor.obj (CochainComplex.single₀ C) Z)))
        (X₂ :=
        ShortComplex.X₂.{v, u}
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
        (ShortComplex.Hom.τ₂
          (Functor.map (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
            ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
                (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
                  (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
              (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
                  Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                    (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0)
                  (Injective.ι Z) ⋯ :
                { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
                  Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                    (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0 }))))
        (ShortComplex.g
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
        ⋯) ∧
    Mono.{v, u} (X :=
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)))
      (Y :=
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
      (ShortComplex.Hom.τ₂
        (Functor.map (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
              (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
                (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
            (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
                Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                  (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0)
                (Injective.ι Z) ⋯ :
              { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
                Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                  (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0 }))))
have zero : Iso (C := ShortComplex.{v, u} C)
  (ShortComplex.mk (X₁ :=
    ShortComplex.X₂.{v, u}
      (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
        (Functor.obj (CochainComplex.single₀ C) Z)))
    (X₂ :=
    ShortComplex.X₂.{v, u}
      (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
    (ShortComplex.Hom.τ₂
      (Functor.map (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
        ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
            (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
              (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
          (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
              Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0)
              (Injective.ι Z) sorry :
            { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
              Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0 }))))
    (ShortComplex.g (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
    sorry)
  (ShortComplex.mk (X₂ := under.{v, u} Z) (X₃ := syzygies.{v, u} (Y := under.{v, u} Z) (Injective.ι Z)) (Injective.ι Z)
    (d (Injective.ι Z)) sorry)
Tactic: refine (ShortComplex.exact_and_mono_f_iff_of_iso ?_).2
          ⟨exact_f_d (Injective.ι Z), by dsimp; infer_instance⟩