Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on p147).

The problem:

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. We can thus regard $C$ as an object in $Fun(B\Bbb Z, Cat)$.

2. We construct the quotient category $C':=C/B\Bbb Z$ by quotienting the morphism space by the $\Bbb Z$ action. This is equivalent to taking the colimit of $C$ as an object in $Fun(B\Bbb Z, Cat)$.

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

The proof in the paper goes as follow.

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

I am rather confused by both the 1st. and 2nd. equivalence. Any comment/reference would be appreciated.

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 $BG$.