/-- If a cospan is the pushout cospan under the initial object, then it is a binary coproduct. -/
def isBinaryCoproductOfIsInitialIsPushout (F : Discrete WalkingPair ⥤ C) (c : Cocone F) {X : C}
    (hX : IsInitial X) (f : X ⟶ F.obj ⟨WalkingPair.left⟩) (g : X ⟶ F.obj ⟨WalkingPair.right⟩)
    (hc :
      IsColimit
        (PushoutCocone.mk (c.ι.app ⟨WalkingPair.left⟩) (c.ι.app ⟨WalkingPair.right⟩ : _) <|
          hX.hom_ext (f ≫ _) (g ≫ _))) :
    IsColimit c
    where
  desc s :=
    hc.desc
      (PushoutCocone.mk (s.ι.app ⟨WalkingPair.left⟩) (s.ι.app ⟨WalkingPair.right⟩) (hX.hom_ext _ _))
  fac s j :=
    Discrete.casesOn j fun j =>
      WalkingPair.casesOn j (hc.fac _ WalkingSpan.left) (hc.fac _ WalkingSpan.right)
  uniq s m J := by
    let c' :=
      PushoutCocone.mk (c.ι.app ⟨WalkingPair.left⟩ ≫ m) (c.ι.app ⟨WalkingPair.right⟩ ≫ m)
        (hX.hom_ext (f ≫ _) (g ≫ _))
    dsimp; rw [← J, ← J]
    apply hc.hom_ext
    rintro (_ | (_ | _)) <;>
      simp only [PushoutCocone.mk_ι_app_zero, PushoutCocone.mk_ι_app, Category.assoc]
    

Proof state:
case none
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F✝ : C ⥤ D
F : Discrete WalkingPair ⥤ C
c : Cocone F
X : C
hX : IsInitial X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left }) (c.ι.app { as := WalkingPair.right })
      (_ : f ≫ c.ι.app { as := WalkingPair.left } = g ≫ c.ι.app { as := WalkingPair.right }))
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
    (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m)
⊢ f ≫ c.ι.app { as := WalkingPair.left } ≫ m =
    f ≫
      c.ι.app { as := WalkingPair.left } ≫
        IsColimit.desc hc
          (PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
            (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m))
case some.left
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F✝ : C ⥤ D
F : Discrete WalkingPair ⥤ C
c : Cocone F
X : C
hX : IsInitial X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left }) (c.ι.app { as := WalkingPair.right })
      (_ : f ≫ c.ι.app { as := WalkingPair.left } = g ≫ c.ι.app { as := WalkingPair.right }))
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
    (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m)
⊢ c.ι.app { as := WalkingPair.left } ≫ m =
    c.ι.app { as := WalkingPair.left } ≫
      IsColimit.desc hc
        (PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
          (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m))
case some.right
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
F✝ : C ⥤ D
F : Discrete WalkingPair ⥤ C
c : Cocone F
X : C
hX : IsInitial X
f : X ⟶ F.obj { as := WalkingPair.left }
g : X ⟶ F.obj { as := WalkingPair.right }
hc :
  IsColimit
    (PushoutCocone.mk (c.ι.app { as := WalkingPair.left }) (c.ι.app { as := WalkingPair.right })
      (_ : f ≫ c.ι.app { as := WalkingPair.left } = g ≫ c.ι.app { as := WalkingPair.right }))
s : Cocone F
m : c.pt ⟶ s.pt
J : ∀ (j : Discrete WalkingPair), c.ι.app j ≫ m = s.ι.app j
c' : PushoutCocone f g :=
  PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
    (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m)
⊢ c.ι.app { as := WalkingPair.right } ≫ m =
    c.ι.app { as := WalkingPair.right } ≫
      IsColimit.desc hc
        (PushoutCocone.mk (c.ι.app { as := WalkingPair.left } ≫ m) (c.ι.app { as := WalkingPair.right } ≫ m)
          (_ : f ≫ c.ι.app { as := WalkingPair.left } ≫ m = g ≫ c.ι.app { as := WalkingPair.right } ≫ m))

Next tactic:
congr 1