homEquivOfIsRightKanExtension — Mathlib

English

There is a natural equivalence between morphisms G ⟶ F' and L ⋙ G ⟶ F, given α and F' with the IsRightKanExtension property.

Русский

Существует естественное эквивалентность между морфизмами G ⇢ F' и L ⋙ G ⇢ F, заданная α и F' с свойством IsRightKanExtension.

LaTeX

$$$ (G \\to F') \\simeq (L \\circ G \\to F) $ (via the canonical hom-set isomorphism).$$

Lean4

/-- If `(F', α)` is a right Kan extension of `F` along `L`, then this
is the induced bijection `(G ⟶ F') ≃ (L ⋙ G ⟶ F)` for all `G`. -/
noncomputable def homEquivOfIsRightKanExtension (G : D ⥤ H) : (G ⟶ F') ≃ (L ⋙ G ⟶ F)
    where
  toFun β := whiskerLeft _ β ≫ α
  invFun β := liftOfIsRightKanExtension _ α _ β
  left_inv β := Functor.hom_ext_of_isRightKanExtension _ α _ _ (by simp)
  right_inv := by cat_disch

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