The Stacks project

Exercise 111.30.12. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$, $K^\bullet$ be complexes of $\text{Fil}^ f(\mathcal{A})$. Assume

1. $K^\bullet$ bounded below and filtered acyclic, and

2. $I^\bullet$ bounded below and consisting of filtered injective objects.

Then any morphism $K^\bullet \to I^\bullet$ is homotopic to zero.