hasSmallLocalizedHom_iff_target — Mathlib

English

Given isomorphisms X ≅ X', Y ≅ Y', HasSmallLocalizedHom W X Y is equivalent to HasSmallLocalizedHom W X' Y'.

Русский

При изоморфизма X ≅ X', Y ≅ Y' имеет место эквивалентность HasSmallLocalizedHom W X Y и HasSmallLocalizedHom W X' Y'.

LaTeX

$$$\text{HasSmallLocalizedHom}(W,X,Y) \iff \text{HasSmallLocalizedHom}(W,X',Y').$$$

Lean4

theorem hasSmallLocalizedHom_iff_target {Y Y' : C} (f : Y ⟶ Y') (hf : W f) :
    HasSmallLocalizedHom.{w} W X Y ↔ HasSmallLocalizedHom.{w} W X Y' :=
  by
  simp only [hasSmallLocalizedHom_iff W W.Q]
  exact small_congr (Iso.homCongr (Iso.refl _) (Localization.isoOfHom W.Q W f hf))

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