Lemma 13.33.8. Let $\mathcal{D}$ be a triangulated category having countable direct sums. Let $\mathcal{A}$ be an abelian category with exact colimits over $\mathbf{N}$. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor commuting with countable direct sums. Then $H(\text{hocolim} K_ n) = \mathop{\mathrm{colim}}\nolimits H(K_ n)$ for any system of objects of $\mathcal{D}$.

**Proof.**
Write $K = \text{hocolim} K_ n$. Apply $H$ to the defining distinguished triangle to get

\[ \bigoplus H(K_ n) \to \bigoplus H(K_ n) \to H(K) \to \bigoplus H(K_ n[1]) \to \bigoplus H(K_ n[1]) \]

where the first map is given by $1 - H(f_ n)$ and the last map is given by $1 - H(f_ n[1])$. Apply Lemma 13.33.6 to see that this proves the lemma. $\square$

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