ScalingStacks

Fact 4.12. Let $F\colon\mathcal{C}\to\mathcal{C}^{\oplus}$ and $G\colon\mathcal{C}\to\mathrm{Kar}(\mathcal{C})$ denote the canonical embeddings of $\mathcal{C}$ into its additive and its idempotent completion, respectively. Then $\mathrm{HH}_{0}(F)$ and $\mathrm{HH}_{0}(G)$ are isomorphisms; see e.g. [4, Sections 3.4 and 3.5].