In ordinary category theory it is a well-known and important fact that the $\hom$ bifunctor into $\text{Set}$ preserves limits. 


I am unable to find a reference for the corresponding fact in infinity category theory nor am I able to write down a proof myself.


I am looking for a statement of the following form:


Let $\mathcal{C}, \mathcal{D}$ be $(\infty, 1)$-categories (considered as a weak Kan complex preferably) and let $F: \mathcal{D} \to \mathcal{C}$ be a diagram. Then, for any object $X \in \mathcal{C}$, we have a natural equivalence between $\lim \hom(X, F(D))$ and $\hom (X, \lim F(D))$ where each $\hom$ is considered as a weak Kan complex.


I would presume that HTT would have a statement to this effect, but I am unable to find one. 



Hom spaces in (∞, 1)-categories
This is [Cisinski](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf), Corollary 6.3.5. The proof is essentially to show that cocontinuous functors out of presheaf $\infty$-categories admit right adjoints given by the usual formula, so that $Hom\_C$ has a left adjoint if $C$ is cocomplete, and then to use the Yoneda embedding for a general $C$.


