Context. I am trying to understand the argument in B.4, THH, p147. It seems to me that the argument boils down to showing :

Let $G$ be a finite group. $f:B'G \rightarrow Spc$ be functor determined by sending the unique point to terminal space $*$ with free $G$-action. Then $$colim_{B'G} f \simeq B(BG)$$

Notational remark:

• $B'G$ is groupoid with one object and morphisms as $G$.
• $B(BG)$ is classifying space of classifying space of $G$.
• Spc we mean the $\infty$-category of spaces, and $colim$ is taken as homotopy colimit

As far as I could find in the literature, we know the case when we have a constant diagram, of which the colimit is given by $BG$. Any comment/reference would be appreciated!