The Stacks project

Lemma 13.26.4. Let $\mathcal{A}$ be an abelian category. Let $u : A \to B$ be a strict monomorphism of $\text{Fil}^ f(\mathcal{A})$ and $f : A \to I$ a morphism from $A$ into a filtered injective object in $\text{Fil}^ f(\mathcal{A})$. Then there exists a morphism $g : B \to I$ such that $f = g \circ u$.

Proof. The pushout $f' : I \to I \amalg _ A B$ of $f$ by $u$ is a strict monomorphism, see Homology, Lemma 12.19.10. Hence the result follows formally from Lemma 13.26.3. $\square$

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