of_domain_equivalence — Mathlib

English

The embedding functor preserves finite limits.

Русский

Функтор вложения сохраняет конечные пределы.

LaTeX

$$$\mathrm{PreservesFiniteLimits}(\mathrm{FreydMitchell.functor}(C))$$$

Lean4

/-- Transport a `HasExactColimitsOfShape` along an equivalence of the shape.

Note: When `C` has finite limits, this lemma holds with the equivalence replaced by a final
functor, see `hasExactColimitsOfShape_of_final` below.
-/
theorem of_domain_equivalence {J J' : Type*} [Category J] [Category J'] (e : J ≌ J') [HasColimitsOfShape J C]
    [HasExactColimitsOfShape J C] :
    haveI : HasColimitsOfShape J' C := hasColimitsOfShape_of_equivalence e
    HasExactColimitsOfShape J' C :=
  haveI : HasColimitsOfShape J' C := hasColimitsOfShape_of_equivalence e
  ⟨preservesFiniteLimits_of_natIso (Functor.Final.colimIso e.functor)⟩

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