English
Variant: for any D : WalkingSpan ⥤ C, a colimit cocone yields an IsPushout on the mapped legs.
Русский
Вариант: для D: WalkingSpan ⥤ C колимит кокон порождает IsPushout на соответствующих рёбрах.
LaTeX
$$$ {D : WalkingSpan ⥤ C} {c : Cocone D} (hc : IsColimit c) : IsPushout (D.map WalkingSpan.Hom.fst) (D.map WalkingSpan.Hom.snd) (c.ι.app .left) (c.ι.app .right) $$$
Lean4
/-- If `c` is a colimiting binary coproduct cocone, and we have an initial object,
then we have `IsPushout 0 0 c.inl c.inr`
(where each `0` is the unique morphism from the initial object). -/
theorem of_is_coproduct {c : BinaryCofan X Y} (h : Limits.IsColimit c) (t : IsInitial Z) :
IsPushout (t.to _) (t.to _) c.inl c.inr :=
of_isColimit
(isPushoutOfIsInitialIsCoproduct _ _ _ _ t
(IsColimit.ofIsoColimit h (Limits.Cocones.ext (Iso.refl c.pt) (by rintro ⟨⟨⟩⟩ <;> simp))))
