functorOp_final — Mathlib

English

For a cofiltered indexed diagram, the functorOp is final.

Русский

Для необобщенного диаграммы, fункторOp финальный.

LaTeX

$$$\mathrm{Profinite.Extend.functorOp}(c) \text{ Final}$$$

Lean4

/-- If the projection maps in the cone are epimorphic and the cone is limiting, then
`Profinite.Extend.functorOp` is final.
-/
theorem functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) :=
  by
  have := functor_initial c hc
  have : ((StructuredArrow.toCostructuredArrow toProfinite c.pt)).IsEquivalence :=
    (inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence)
  exact Functor.final_comp (functor c).op _

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