of_final — Mathlib

English

If T is an initial functor, then CostructuredArrow.pre T S X is initial for any S and X.

Русский

Если T начальная функтор, то CostructuredArrow.pre T S X начальная для любых S и X.

LaTeX

$$$\operatorname{Initial}(T) \Rightarrow \operatorname{Initial}(\operatorname{CostructuredArrow.pre}(T,S,X))$$$

Lean4

/-- Final functors preserve filteredness.

This can be seen as a generalization of `IsFiltered.of_right_adjoint` (which states that right
adjoints preserve filteredness), as right adjoints are always final, see `final_of_adjunction`.
-/
theorem of_final (F : C ⥤ D) [Final F] [IsFiltered C] : IsFiltered D :=
  { IsFilteredOrEmpty.of_final F with nonempty := Nonempty.map F.obj IsFiltered.nonempty }

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