7:03 PM
@DylanWilson OK, I guess I understand. I'd say things like this. Let $\mathcal{X}$ be an infinity-topos, and $B$ an object of it. Then $B$ can be thought of as an "internal $\infty$-groupoid" of $\mathcal{X}$, by abstract abstract nonsense. The infinity-topos has an internal model of itself (up to size issues), an $\infty$-category object in $\mathcal{X}$ I'll call $\Omega$. ...
...Then there's an $\infty$-category $[B,\Omega]$ of "internal functors" and we have $[B,\Omega]=\mathcal{X}_{/B}$, basically formally using descent.
Then the diagonal map $B\to B\times B$, viewed as a morphism in $\mathcal{X}_{/B}$, corresponds to a natural transformation of functors in $[B,\Omega]$.
Taking $B=B_G\Gamma$ gives what you say, although some features of this are not formal consequences of the formalism I described: e.g., that $(\mathrm{Spaces}_G)_{B_G\Gamma}$ is equivalent to presheaves on certain $G\times \Gamma$-orbits.
Given $G$ and $\Gamma$, there is a kind of candidate for $B_G\Gamma$, which I'll call $B_G^{\mathrm{borel}} \Gamma$. It is defined as $G$-space as $\mathrm{Map}(EG, B\Gamma)$. In some cases (e.g., if $\Gamma$ is finite or a torus) it is the same as $B_G\Gamma$.
If not, then $(\mathrm{Spaces}_G)_{/B_G^{\mathrm{borel}}\Gamma}$ is equivalent to presheaves on something complicated and not describable using Lie groups, probably.