The Stacks project

Lemma 13.26.2. Let $\mathcal{A}$ be an abelian category. An object $I$ of $\text{Fil}^ f(\mathcal{A})$ is filtered injective if and only if there exist $a \leq b$, injective objects $I_ n$, $a \leq n \leq b$ of $\mathcal{A}$ and an isomorphism $I \cong \bigoplus _{a \leq n \leq b} I_ n$ such that $F^ pI = \bigoplus _{n \geq p} I_ n$.

Proof. Follows from the fact that any injection $J \to M$ of $\mathcal{A}$ is split if $J$ is an injective object. Details omitted. $\square$

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