Lemma 12.29.5. 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 $v$,

$v$ transforms injective maps into injective maps,

$\mathcal{A}$ has enough injectives,

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

$\mathcal{A}$ has functorial injective hulls.

Then $\mathcal{B}$ has functorial injective hulls.

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