Let $p:I\to Cat_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x_i,y_i\in C_i=p(i)$ be the corresponding objects in $C_i$. Then we have a new diagram $q_{x,y}: I\to \mathcal{S}$ given by $q_{x,y}(i)=Map_{C_i}(x_i,y_i)$, where $\mathcal{S}$ is the infinity category of spaces.
Question: Is it always true that $Map_C(x,y)\simeq\lim q_{x,y}$?
Yes. To see this, let us make the preliminary observation that it suffices to prove that this holds for products and pullbacks since we can decompose a general limit into these two special cases.
Let us begin with products. If $I$ is discrete then we want to show that $\prod_i \hom_{\mathcal{C_i}}(x_i,y_i) \cong \hom_{\prod_i \mathcal{C_i}}((x_i)_{i \in I},(y_i)_{i \in I})$. To see this, note that we can realize the limit as the honest product of quasicategories in simplicial sets. By considering the model of $\hom_{\mathcal{D}}(d_0,d_1)$ given by $\mathbf{1} \times_{\mathcal{D}} \mathcal{D}^{\Delta^1} \times_{\mathcal{D}} \mathbf{1}$ the conclusion follows immediately as the two $\infty$-groupoids admit isomorphic models as simplicial sets.
Next, we want to show that the same is true for pullbacks. Again, we can rectify our diagram, this time into cospan of $\infty$-categories $\mathcal{C}_0 \to \mathcal{C}_{01} \leftarrow \mathcal{C}_1$. We can replace $\mathcal{C}_0$ and $\mathcal{C}_0 \to \mathcal{C}_{01}$ such that the latter becomes a fibration by changing $\mathcal{C}_0$ up to equivalence. In this case, the desired homotopy limit is realized by the actual pullback and exactly the same argument as for products applies (exponentials and limits commute with limits in simplicial sets).
