Definition 12.27.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has *functorial injective embeddings* if there exists a functor

such that

$s \circ J = \text{id}_\mathcal {A}$,

for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $J(A)$ is injective, and

for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $t(J(A))$ is an injective object of $\mathcal{A}$.

We will denote such a functor by $A \mapsto (A \to J(A))$.

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