ScalingStacks

Fact 4.11. Any functor $F\colon\mathcal{C}\to\mathcal{D}$ of $K$-linear categories induces natural $K$-module homomorphism $\mathrm{HH}_{0}(F)\colon\mathrm{HH}_{0}(\mathcal{C})\to\mathrm{HH}_{0}(\mathcal{D})$ sending $[f\colon x\to x]\mapsto[F(f)\colon F(x)\to F(x)]]$. This is well-defined since $f\circ g-g\circ f\mapsto F(f)\circ F(g)-F(g)\circ F(f)$. If $F$ is an equivalence, then $\mathrm{HH}_{0}(F)$ is an isomorphism; see e.g. [4].