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

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

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

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

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

5. $\mathcal{A}$ has functorial injective embeddings.

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

Proof. Let $A \mapsto (A \to J(A))$ be a functorial injective embedding on $\mathcal{A}$. Then $B \mapsto (B \to uJ(vB))$ is a functorial injective embedding on $\mathcal{B}$. Compare with the proof of Lemma 12.29.3. $\square$

Comment #8494 by Laurent Moret-Bailly on

:I have searched for the definition of "injective hull". It does not appear until 08Y2 (currently 47.5.1), and then only as an ad hoc version for modules.

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