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

The question to be solved:

1. We begin with category $C$ with a free $\Bbb Z$-action which fixes the objects of $C$. I.e. Each $g \in \Bbb Z$ induces a functor $$g:C \rightarrow C $$ that fixes the objects.

2. We construct the quotient category $C':=C/B\Bbb Z$ by quotienting the morphism space by the $\Bbb Z$ action.

3. We wish to compute $|BC'|$. What we do know is that $|BC|\simeq *$.

The argument in the proof goes as follow.

$$|BC'| \simeq |BC|/B\Bbb Z \simeq B(B\Bbb Z) $$

I am rather confused by both the 1st. and 2nd. equivalence.

My confusions.

For the 1st. It seems to me the author claims we have a commuting diagram $$ Fun(B\Bbb Z,Cat) \rightarrow Cat $$ $$Fun(B\Bbb Z, Spc) \rightarrow Spc $$ where we take colimits horizontally and geometric realization $|N(-)|$ vertically.

For the 2nd. 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!