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

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

2. $v$ 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. $\square$

Comment #5064 by Remy on

Suggested slogan: functors with an exact left adjoint preserve injectives.

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