injective_of_adjoint — Mathlib

English

The preadditive Yoneda functor is faithful.

Русский

Функтор пр предадитивной Яノеда ремесленничество достоверен (верен).

LaTeX

$$$\text{preadditiveYoneda}$ is faithful$$

Lean4

theorem injective_of_adjoint (adj : L ⊣ R) (J : D) [Injective J] : Injective <| R.obj J :=
  ⟨fun {A} {_} g f im =>
    ⟨adj.homEquiv _ _ (factorThru ((adj.homEquiv A J).symm g) (L.map f)),
      (adj.homEquiv _ _).symm.injective (by simp [Adjunction.homEquiv_unit, Adjunction.homEquiv_counit])⟩⟩

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