** A functor with an exact left adjoint preserves injectives **

Lemma 12.29.1. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume

$u$ is right adjoint to $v$, and

$v$ transforms injective maps into injective maps.

Then $u$ transforms injectives into injectives.

**Proof.**
Let $I$ be an injective object of $\mathcal{A}$. Let $\varphi : N \to M$ be an injective map in $\mathcal{B}$ and let $\alpha : N \to uI$ be a morphism. By adjointness we get a morphism $\alpha : vN \to I$ and by assumption $v\varphi : vN \to vM$ is injective. Hence as $I$ is an injective object we get a morphism $\beta : vM \to I$ extending $\alpha $. By adjointness again this corresponds to a morphism $\beta : M \to uI$ as desired.
$\square$

## Comments (1)

Comment #5064 by Remy on