Let $\bf Top$ a convenient category of topological spaces, $G$ a group in $\bf Top$, ${}^G\bf Top$ the category of (left) $G$-spaces, and $Sgrp(G)$ the poset of subgroups of $G$.

Define two functors [and prepare yourself to a couple of slight abuses of notation]:

- ${}^G{\bf Top}\times Sgrp(G)^\text{op}\to {\bf Top}\colon (X,H)\mapsto X^H$, sending a space into the subspace of fixed point of the $H$ action on $X$;
- $Sgrp(G)\to \mathbf{Top}\colon H\mapsto G/H$ with the induced topology.

Now fix a space $X\in \bf Top$ and consider the functor $Sgrp(G)^\text{op}\times Sgrp(G)\to \bf Top$ defined by $(H,K)\mapsto X^H \times G/K$.

How can you prove (if it is true) that
$$
\int^{H\in Sgrp(G)} X^H\times G/H\cong X?
$$
purely *coendy* proofs are welcome.