ScalingStacks

Lemma 3.5 ([15], Lemma 2.8). Suppose $S$ is a compact object of a triangulated category $𝒯$ and we have a sequence of objects and morphisms of $𝒯$:

$$X_0\to X_1\to X_2\to X_3\to \mathrm{\cdots }$$

then $\text{colim}(\text{Hom}(S,X_n))\cong \text{Hom}(S,\text{hocolim}{X}_n)$.