Lemma 12.27.3. Let $\mathcal{A}$ be an abelian category. Suppose $I_\omega $, $\omega \in \Omega $ is a set of injective objects of $\mathcal{A}$. If $\prod _{\omega \in \Omega } I_\omega $ exists then it is injective.

**Proof.**
Omitted.
$\square$

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