ScalingStacks

Proof. It is clear from the construction of the trace described above that it descends to a natural transformation of functors of $\infty$-categories $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH$. One checks on each side that the trace commutes with filtered homotopy colimits of spectral categories, and so the result follows. ∎