Lemma 15.55.9. Let $R$ be a ring. The construction above defines a covariant functor $M \mapsto (M \to J(M))$ from the category of $R$-modules to the category of arrows of $R$-modules such that for every module $M$ the output $M \to J(M)$ is an injective map of $M$ into an injective $R$-module $J(M)$.

**Proof.**
Follows from the above.
$\square$

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