theorem binaryCofan_isColimit_iff {X Y : TopCat} (c : BinaryCofan X Y) :
    Nonempty (IsColimit c) ↔
      OpenEmbedding c.inl ∧ OpenEmbedding c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr) := by
  classical
    constructor
    · rintro ⟨h⟩
      rw [← show _ = c.inl from
          h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.left⟩,
        ← show _ = c.inr from
          h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.right⟩]
      dsimp
      refine' ⟨(homeoOfIso <| h.coconePointUniqueUpToIso
        (binaryCofanIsColimit X Y)).symm.openEmbedding.comp openEmbedding_inl,
          (homeoOfIso <| h.coconePointUniqueUpToIso
            (binaryCofanIsColimit X Y)).symm.openEmbedding.comp openEmbedding_inr, _⟩
      erw [Set.range_comp, ← eq_compl_iff_isCompl, coe_comp, coe_comp, Set.range_comp _ Sum.inr,
        ← Set.image_compl_eq (homeoOfIso <| h.coconePointUniqueUpToIso
            (binaryCofanIsColimit X Y)).symm.bijective, Set.compl_range_inr, Set.image_comp]
      aesop
    · rintro ⟨h₁, h₂, h₃⟩
      have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by
        rw [eq_compl_iff_isCompl.mpr h₃.symm]
        exact fun _ => or_not
      refine' ⟨BinaryCofan.IsColimit.mk _ _ _ _ _⟩
      · intro T f g
        refine' ContinuousMap.mk _ _
        · exact fun x =>
            if h : x ∈ Set.range c.inl then f ((Equiv.ofInjective _ h₁.inj).symm ⟨x, h⟩)
            else g ((Equiv.ofInjective _ h₂.inj).symm ⟨x, (this x).resolve_left h⟩)
        rw [continuous_iff_continuousAt]
        intro x
        by_cases h : x ∈ Set.range c.inl
        · revert h x
          apply (IsOpen.continuousOn_iff _).mp
          · rw [continuousOn_iff_continuous_restrict]
            convert_to Continuous (f ∘ (Homeomorph.ofEmbedding _ h₁.toEmbedding).symm)
            · ext ⟨x, hx⟩
              exact dif_pos hx
            apply Continuous.comp
            · exact f.continuous_toFun
            · continuity
          · exact h₁.open_range
        · revert h x
          apply (IsOpen.continuousOn_iff _).mp
          · rw [continuousOn_iff_continuous_restrict]
            have : ∀ a, a ∉ Set.range c.inl → a ∈ Set.range c.inr := by
              rintro a (h : a ∈ (Set.range c.inl)ᶜ)
              rwa [eq_compl_iff_isCompl.mpr h₃.symm]
            convert_to Continuous
                (g ∘ (Homeomorph.ofEmbedding _ h₂.toEmbedding).symm ∘ Subtype.map _ this)
            · ext ⟨x, hx⟩
              exact dif_neg hx
            apply Continuous.comp
            · exact g.continuous_toFun
            · apply Continuous.comp
              · continuity
              · rw [embedding_subtype_val.toInducing.continuous_iff]
                exact continuous_subtype_val
          · change IsOpen (Set.range c.inl)ᶜ
            rw [← eq_compl_iff_isCompl.mpr h₃.symm]
            exact h₂.open_range
      · intro T f g
        ext x
        refine' (dif_pos _).trans _
        · exact ⟨x, rfl⟩
        · dsimp; conv_lhs => erw [Equiv.ofInjective_symm_apply]
      · intro T f g
        ext x
        refine' (dif_neg _).trans _
        · rintro ⟨y, e⟩
          have : c.inr x ∈ Set.range c.inl ⊓ Set.range c.inr := ⟨⟨_, e⟩, ⟨_, rfl⟩⟩
          rwa [disjoint_iff.mp h₃.1] at this
        · exact congr_arg g (Equiv.ofInjective_symm_apply _ _)
      · rintro T _ _ m rfl rfl
        ext x
        change m x = dite _ _ _
        split_ifs <;> 

Proof state:
case pos
J : Type v
inst✝ : SmallCategory J
X Y : TopCat
c : BinaryCofan X Y
h₁ : OpenEmbedding ⇑(BinaryCofan.inl c)
h₂ : OpenEmbedding ⇑(BinaryCofan.inr c)
h₃ : IsCompl (Set.range ⇑(BinaryCofan.inl c)) (Set.range ⇑(BinaryCofan.inr c))
this :
  ∀ (x : (forget TopCat).obj (((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left })),
    x ∈ Set.range ⇑(BinaryCofan.inl c) ∨ x ∈ Set.range ⇑(BinaryCofan.inr c)
T : TopCat
m : c.pt ⟶ T
x : (forget TopCat).obj c.pt
h✝ : x ∈ Set.range ⇑(BinaryCofan.inl c)
⊢ m x =
    (fun h =>
        (BinaryCofan.inl c ≫ m)
          ((Equiv.ofInjective ⇑(BinaryCofan.inl c) (_ : Function.Injective ⇑(BinaryCofan.inl c))).symm
            { val := x, property := h }))
      h✝

Next tactic:
exact congr_arg _ (Equiv.apply_ofInjective_symm _ ⟨_, _⟩).symm