case h.e'_5.h
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
e_2✝ : (forget TopCat).obj Y = ↑Y
⊢ (forget TopCat).map pullback.snd =
    ↑(homeoOfIso (asIso pullback.snd)) ∘
      (forget TopCat).map (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g))
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
⊢ Embedding ((forget TopCat).map pullback.snd)
convert (homeoOfIso (asIso (pullback.snd : pullback (𝟙 S) g ⟶  _))).embedding.comp
    (pullback_map_embedding_of_embeddings (i₂ := 𝟙 Y)
      f g (𝟙 S) g H (homeoOfIso (Iso.refl _)).embedding (𝟙 _) rfl (by simp))
case h.e'_5.h
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
e_2✝ : (forget TopCat).obj Y = ↑Y
⊢ (forget TopCat).map pullback.snd =
    (forget TopCat).map
      (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫
        (asIso pullback.snd).hom)
case h.e'_5.h
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
e_2✝ : (forget TopCat).obj Y = ↑Y
⊢ (forget TopCat).map pullback.snd =
    ↑(homeoOfIso (asIso pullback.snd)) ∘
      (forget TopCat).map (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g))
erw [← coe_comp]
no goals
case h.e'_5.h
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
e_2✝ : (forget TopCat).obj Y = ↑Y
⊢ (forget TopCat).map pullback.snd =
    (forget TopCat).map
      (pullback.map f g (𝟙 S) g f (𝟙 Y) (𝟙 S) (_ : f ≫ 𝟙 S = f ≫ 𝟙 S) (_ : g ≫ 𝟙 S = 𝟙 Y ≫ g) ≫
        (asIso pullback.snd).hom)
simp
no goals
J : Type v
inst✝ : SmallCategory J
X✝ Y✝ Z : TopCat
X Y S : TopCat
f : X ⟶ S
H : Embedding ((forget TopCat).map f)
g : Y ⟶ S
⊢ g ≫ 𝟙 S = 𝟙 Y ≫ g
simp