Lemma 13.13.3. Let $\mathcal{A}$ be an abelian category.
The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$, $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet ))$ is homological.
The functor $K(\text{Fil}^ f(\mathcal{A})) \rightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0(\text{gr}^ p(K^\bullet ))$ is homological.
The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0((\text{forget }F)K^\bullet )$ is homological.
Proof. This follows from the fact that $H^0 : K(\mathcal{A}) \to \mathcal{A}$ is homological, see Lemma 13.11.1 and the fact that the functors $\text{gr}, \text{gr}^ p, (\text{forget }F)$ are exact functors of triangulated categories. See Lemma 13.4.20. $\square$
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