**Some background and notation**


Let $Sh\_{\infty}(Cartsp)$ be the infinity category of smooth simplicial sheaves on the site of cartesian spaces (convex open subsets of $\mathbb{R}^n$ and smooth maps between them), equipped with the topology of good open covers (contractible finite intersections). 


This infinity topos is cohesive and in particular the functor ${\rm disc}:\infty-\mathscr{G}{\rm rpd}\to Sh\_{\infty}(Cartsp)$ admits an $\infty$ left adjoint $\Pi$ which preserves finite products. This is the functor I am calling geometric realization.


**Question**


Let me denote the internal mapping simplicial sheaf, coming from the cartesian closed structure on $Sh\_{\infty}(Cartsp)$, by $[M,X]$. Given a smooth compact manifold $M$ and a smooth simplicial sheaf $X$, I would like to compare the geometric realization of $[M,X]$ and the mapping space of the geometric realizations of $M$ and $X$ respectively. Ideally, I would hope for an equivalence 
$$\Pi[M,X]\simeq Map(\Pi(M),\Pi(X))\;.$$
Is this true? I haven't been able to come up with a counter example and I know this is true if one takes sheaves with values in a *stable* infinity category (there one can use the fact that arbitrary colimits commute with finite limits along with descent with respect to a finite cover of $M$). Any help here would be greatly appreciated!



Geometric realization of the mapping stack
Yes. This question was already asked and answered on the nForum: <https://nforum.ncatlab.org/discussion/6816/the-shape-of-function-objects/>


