Exercise 111.30.10. Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. Let $K^\bullet $ be a bounded below complex of objects of $\text{Fil}^ f(\mathcal{A})$. Show there exists a filtered quasi-isomorphism $\alpha : K^\bullet \to I^\bullet $ with $I^\bullet $ a complex of $\text{Fil}^ f(\mathcal{A})$ having filtered injective terms $I^ n$, and bounded below. In fact, we may choose $\alpha $ such that each $\alpha ^ n$ is a strict monomorphism.

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