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.