The Stacks project

Lemma 12.29.3. 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 $v$,

  2. $v$ transforms injective maps into injective maps,

  3. $\mathcal{A}$ has enough injectives, and

  4. $vB = 0$ implies $B = 0$ for any $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

Then $\mathcal{B}$ has enough injectives.

Proof. Pick $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$. Pick an injection $vB \to I$ for $I$ an injective object of $\mathcal{A}$. According to Lemma 12.29.1 and the assumptions the corresponding map $B \to uI$ is the injection of $B$ into an injective object. $\square$

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