The Stacks project

Exercise 111.30.9. Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. Let $A$ be an object of $\text{Fil}^ f(\mathcal{A})$. Show there exists a complex $I^\bullet $ of $\text{Fil}^ f(\mathcal{A})$, and a morphism $A[0] \to I^\bullet $ such that

1. each $I^ p$ is filtered injective,

2. $I^ p = 0$ for $p < 0$, and

3. $A[0] \to I^\bullet $ is a filtered quasi-isomorphism.