/-- The pushout of two epimorphisms exists. -/
@[irreducible, nolint defLemma] -- Porting note: made a def and re-added irreducible
def pushout_of_epi {X Y Z : C} (a : X ⟶ Y) (b : X ⟶ Z) [Epi a] [Epi b] :
    HasColimit (span a b) :=
  let ⟨P, f, hfa, i⟩ := normalEpiOfEpi a
  let ⟨Q, g, hgb, i'⟩ := normalEpiOfEpi b
  let ⟨a', ha'⟩ :=
    CokernelCofork.IsColimit.desc' i (cokernel.π (coprod.desc f g)) <|
      calc
        f ≫ cokernel.π (coprod.desc f g) =
            coprod.inl ≫ coprod.desc f g ≫ cokernel.π (coprod.desc f g) :=
          by rw [coprod.inl_desc_assoc]
        _ = coprod.inl ≫ (0 : P ⨿ Q ⟶ cokernel (coprod.desc f g)) := by rw [cokernel.condition]
        _ = 0 := HasZeroMorphisms.comp_zero _ _

  let ⟨b', hb'⟩ :=
    CokernelCofork.IsColimit.desc' i' (cokernel.π (coprod.desc f g)) <|
      calc
        g ≫ cokernel.π (coprod.desc f g) =
            coprod.inr ≫ coprod.desc f g ≫ cokernel.π (coprod.desc f g) :=
          by rw [coprod.inr_desc_assoc]
        _ = coprod.inr ≫ (0 : P ⨿ Q ⟶ cokernel (coprod.desc f g)) := by rw [cokernel.condition]
        _ = 0 := HasZeroMorphisms.comp_zero _ _

  HasColimit.mk
    { cocone :=
        PushoutCocone.mk a' b' <| by
          simp only [Cofork.π_ofπ] at ha' hb'
          rw [ha', hb']
      isColimit :=
        PushoutCocone.IsColimit.mk _
          (fun s =>
            cokernel.desc (coprod.desc f g) (b ≫ PushoutCocone.inr s) <|
              coprod.hom_ext
                (calc
                  coprod.inl ≫ coprod.desc f g ≫ b ≫ PushoutCocone.inr s =
                      f ≫ b ≫ PushoutCocone.inr s :=
                    by rw [coprod.inl_desc_assoc]
                  _ = f ≫ a ≫ PushoutCocone.inl s := by rw [PushoutCocone.condition]
                  _ = 0 ≫ PushoutCocone.inl s := by rw [← Category.assoc, eq_whisker hfa]
                  _ = coprod.inl ≫ 0 := by rw [comp_zero, zero_comp]
                  )
                (calc
                  coprod.inr ≫ coprod.desc f g ≫ b ≫ PushoutCocone.inr s =
                      g ≫ b ≫ PushoutCocone.inr s :=
                    by rw [coprod.inr_desc_assoc]
                  _ = 0 ≫ PushoutCocone.inr s := by rw [← Category.assoc, eq_whisker hgb]
                  _ = coprod.inr ≫ 0 := by rw [comp_zero, zero_comp]
                  ))
          (fun s =>
            (cancel_epi a).1 <| by
              rw [CokernelCofork.π_ofπ] at ha'
              have reassoced {W : C} (h : cokernel (coprod.desc f g) ⟶ W) : a ≫ a' ≫ h
                = cokernel.π (coprod.desc f g) ≫ h := by rw [← Category.assoc, eq_whisker ha']
              

Proof state:
C : Type u_1
inst✝⁶ : Category.{?u.142904, u_1} C
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasFiniteCoproducts C
inst✝³ : HasCokernels C
inst✝² : NormalEpiCategory C
X Y Z : C
a : X ⟶ Y
b : X ⟶ Z
inst✝¹ : Epi a
inst✝ : Epi b
P : C
f : P ⟶ X
hfa : f ≫ a = 0
i : IsColimit (CokernelCofork.ofπ a hfa)
Q : C
g : Q ⟶ X
hgb : g ≫ b = 0
i' : IsColimit (CokernelCofork.ofπ b hgb)
a' : (CokernelCofork.ofπ a hfa).pt ⟶ cokernel (coprod.desc f g)
ha' : a ≫ a' = cokernel.π (coprod.desc f g)
b' : (CokernelCofork.ofπ b hgb).pt ⟶ cokernel (coprod.desc f g)
hb' : Cofork.π (CokernelCofork.ofπ b hgb) ≫ b' = cokernel.π (coprod.desc f g)
s : PushoutCocone a b
reassoced : ∀ {W : C} (h : cokernel (coprod.desc f g) ⟶ W), a ≫ a' ≫ h = cokernel.π (coprod.desc f g) ≫ h
⊢ a ≫
      a' ≫
        (fun s =>
            cokernel.desc (coprod.desc f g) (b ≫ PushoutCocone.inr s)
              (_ : coprod.desc f g ≫ b ≫ PushoutCocone.inr s = 0))
          s =
    a ≫ PushoutCocone.inl s

Next tactic:
simp [reassoced , PushoutCocone.condition s]