Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}\_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure on $\mathrm{Set}\_\Delta$ to define this). It has been proved that the $(\infty,1)$-category 2-$\mathrm{Cat}$ of $(\infty,2)$-categories is Cartesian closed. The category of bisimplicial sets $\mathrm{Fun}(\Delta^{op}, \mathrm{Set}\_\Delta)$ is also Cartesian closed. 


My question is:


Does $Y^X$ in 2-$\mathrm{Cat}$ coincide with $Y^X$ in $\mathrm{Fun}(\Delta^{op}, \mathrm{Set}\_\Delta)$, i.e. is the latter one fibrant?


If not, is there a reference about the relation between the two (more specific than saying the first $Y^X$ is a fibrant replacement of the latter $Y^X$)?



Internal hom in $(\infty,2)$-categories
Yes because the model structure on Fun(Δ^op, sSet)
is a cartesian model structure.
This follows from the fact that the Joyal model structure is cartesian,
and §10, §11 in Rezk's paper “A model for the homotopy theory of homotopy theory”.


