Lemma 13.4.20. Let $\mathcal{D}$ be a pre-triangulated category. Let $\mathcal{A}$ be an abelian category. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor.

1. Let $\mathcal{D}'$ be a pre-triangulated category. Let $F : \mathcal{D}' \to \mathcal{D}$ be an exact functor. Then the composition $H \circ F$ is a homological functor as well.

2. Let $\mathcal{A}'$ be an abelian category. Let $G : \mathcal{A} \to \mathcal{A}'$ be an exact functor. Then $G \circ H$ is a homological functor as well.

Proof. Omitted. $\square$

Comment #1290 by jpg on

In (2) "Hence" is repeated instead of "Let", "then"

Comment #3048 by Chung Ching on

In (1), it should be $H\circ F$ instead of $G\circ F$.

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• 13 comment(s) on Section 13.4: Elementary results on triangulated categories

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