final_fst — Mathlib

English

In CostructuredArrow, if C is cofilted, T initial, S initial, and morphism u given, then the map₂ construction is initial.

Русский

В CostructuredArrow, при cofiltro инициализированных T и S и заданном гомоморфизме u, map₂ начален.

LaTeX

$$Same as 66644 but repeated: [IsCofiltered(C)] (T: C ⥤ D) [T.Initial] (S: D ⥤ E) [S.Initial] (d : D) (e : E) (u : S.obj d ⟶ e) : Initial (map₂ (F := 𝟭 _) (U := T ⋙ S) (\text{1}_{T ⋙ S}) u)$$

Lean4

instance final_fst [IsFilteredOrEmpty C] [IsFiltered D] : (Prod.fst C D).Final :=
  by
  apply Functor.final_of_exists_of_isFiltered
  · exact fun c => ⟨(c, nonempty.some), ⟨𝟙 c⟩⟩
  · intro c ⟨c', d'⟩ f g
    exact ⟨(coeq f g, d'), (coeqHom f g, 𝟙 d'), coeq_condition _ _⟩

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