This is rather specific  B.5 of Thomas Nikolaus, Peter Scholze, _On topological cyclic homology_, arXiv:[1707.01799][1] (on last line  p147), which I am having fundamental confusion. 


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We have the categories $\Lambda:=\Lambda_\infty/B\Bbb Z,  \Lambda_\infty$ explained in my previous [question][1]. 

In B.5, the authors describes a functor given by composition 
$$ Fun(\Lambda^{op}, C) \rightarrow Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C) \rightarrow Fun^{B\Bbb Z}(pt, C) = C^{BB\Bbb Z} =  C^{B\Bbb T} $$

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I have fundamental confusion in the first 2 arrows. 

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**Q : What exactly is the cateogry $Fun^{B\Bbb Z}(\Lambda_\infty^{op}, C)$**. I understand it is to be understood as $B\Bbb Z$ equivariant maps. 

But how is this made precise? 

Irregardless, any definition should be that  we have a subcat. 
$$ Fun^{B\Bbb Z}(\Lambda_\infty, C) \subset Fun(\Lambda_\infty^{op}, C )$$

But without a concrete meaning, I could not make sense of the following two. 

**Q2 why does taking collimit preserve $B\Bbb Z$-equivariance?** 

**Q3: How do we show $Fun^{B\Bbb Z}(pt, C)=C^{BB\Bbb Z}$?** 

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I suppose we could  replace $\Bbb Z$ by any topological group $G$.


  [1]: https://mathoverflow.net/questions/376490/computing-homotopy-colimit-of-a-space-with-free-s1-action