This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my understanding precise.

Particular goal:

How the Tate diagonal map (III.1 of paper) map $T_p: Sp \rightarrow Sp$ $$ X \mapsto (X \otimes \cdots \otimes X)^{tC_p}$$ is defined rigorously. What I could define: I could define a map $$ Sp \rightarrow Sp^{\times n} \rightarrow Sp$$ $$ X \mapsto (X,\ldots, X) \mapsto X \otimes \cdots \otimes X $$ using monoidal structure of $Sp^\otimes$ of spectra.

Question How do I rigorously lift this to map $Sp^{BC_p}$? ( allowing me to apply the Tate functor $(-)^{tC_p} : Sp^{BC_p} \rightarrow Sp$. )

EDIT: Most of my question of this goal have been resolved in the replies below (of which all are nice answers). I still have the following confusion

how does one prove formulas fo the underlying(under the notation of Maxime) the adjunction: $$ U:Sp^{BG} \rightarrow Sp:Ind, CoInd $$ of "forgetful"/"inclusion"? where Ind and CoInd are left and right adjoint respectively. i.e. It seems that $$ \bigoplus_g X \simeq UInd X $$ $$ U CoInd X \simeq ?? $$

In particular I am confused about the computation $CoInd(Sp) \simeq Sp^{\times n}$.

This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my understanding precise.

Particular goal:

How the map (III.1) $T_p: Sp \rightarrow Sp$ $$ X \mapsto (X \otimes \cdots \otimes X)^{tC_p}$$ is defined rigorously. What I could define: I could define a map $$ Sp \rightarrow Sp^{\times n} \rightarrow Sp$$ $$ X \mapsto (X,\ldots, X) \mapsto X \otimes \cdots \otimes X $$ using monoidal structure of $Sp^\otimes$ of spectra.

Question How do I rigorously lift this to map $Sp^{BC_p}$? ( allowing me to apply the Tate functor $(-)^{tC_p} : Sp^{BC_p} \rightarrow Sp$. )

EDIT: Most of my question of this goal have been resolved in the replies below (of which all are nice answers). I still have the following confusion

how does one prove formulas fo the underlying(under the notation of Maxime) the adjunction: $$ U:Sp^{BG} \rightarrow Sp:Ind, CoInd $$ of "forgetful"/"inclusion"? where Ind and CoInd are left and right adjoint respectively. i.e. It seems that $$ \bigoplus_g X \simeq UInd X $$ $$ U CoInd X \simeq ?? $$

In particular I am confused about the computation $CoInd(Sp) \simeq Sp^{\times n}$.

This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my understanding precise.

Particular goal:

How the Tate diagonal map (III.1 of paper) map $T_p: Sp \rightarrow Sp$ $$ X \mapsto (X \otimes \cdots \otimes X)^{tC_p}$$ is defined rigorously. What I could define: I could define a map $$ Sp \rightarrow Sp^{\times n} \rightarrow Sp$$ $$ X \mapsto (X,\ldots, X) \mapsto X \otimes \cdots \otimes X $$ using monoidal structure of $Sp^\otimes$ of spectra.

Question How do I rigorously lift this to map $Sp^{BC_p}$? ( allowing me to apply the Tate functor $(-)^{tC_p} : Sp^{BC_p} \rightarrow Sp$. )

Thought so far:

1. I think the approach that I look forward to utilizes the UP , $BC_p = *_{hG}$, where $*$ is the one point category in $Cat$ (perhaps some kind of Kan extensiono) where we are using the adjunction $$ colim: Cat^{BG} \rightarrow Cat$$

2. In otherwords now that I have defined a functor $T_p \in Fun(Sp,Sp)$ I simply Left Kan extend along $* \rightarrow *_{hG}$, yielding $$ l_! T_p: Fun(BG,Fun(Sp,SP)) $$ It does seem to me that intuitively $l_!T_p(*) \simeq X \otimes \cdots \otimes X$ but I couldn't make this precise.

3. Side question. is it true that inclusion $Spc \rightarrow Cat$ commutes with all co/limits? It is right adjoint taking underlying groupoid, but is also seems to be a left adjoiont, where we "freely invert all morphisms."

This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my understanding precise.

Particular goal:

How the Tate diagonal map (III.1 of paper) map $T_p: Sp \rightarrow Sp$ $$ X \mapsto (X \otimes \cdots \otimes X)^{tC_p}$$ is defined rigorously. What I could define: I could define a map $$ Sp \rightarrow Sp^{\times n} \rightarrow Sp$$ $$ X \mapsto (X,\ldots, X) \mapsto X \otimes \cdots \otimes X $$ using monoidal structure of $Sp^\otimes$ of spectra.

Question How do I rigorously lift this to map $Sp^{BC_p}$? ( allowing me to apply the Tate functor $(-)^{tC_p} : Sp^{BC_p} \rightarrow Sp$. )

EDIT: Most of my question of this goal have been resolved in the replies below (of which all are nice answers). I still have the following confusion

how does one prove formulas fo the underlying(under the notation of Maxime) the adjunction: $$ U:Sp^{BG} \rightarrow Sp:Ind, CoInd $$ of "forgetful"/"inclusion"? where Ind and CoInd are left and right adjoint respectively. i.e. It seems that $$ \bigoplus_g X \simeq UInd X $$ $$ U CoInd X \simeq ?? $$

In particular I am confused about the computation $CoInd(Sp) \simeq Sp^{\times n}$.