Complete Segal spaces and quasi-categories are two common models for the theory of $(\infty,1)$-categories, and both are equipped with a natural notion of hom spaces. For complete Segal spaces, which in particular are simplicial spaces $X\colon \Delta^{\rm op} \to \mathcal{S}$, we think of $X_0$ as the space of object, $X_1$ as the space of morphism &c. Thus if $x,y\in X_{0,0}$, then the space of homs should be $x\times^h_{X_0} X_1 \times^h_{X_0} y$.

On the other hand, if $\mathcal{C}$ is a quasi-category, and $x,y\in \mathcal{C}_0$ are objects in $\mathcal{C}$, then the mapping space from $x$ to $y$ is defined (for example, there are other versions) by the rule ${\rm Map}_\mathcal{C}(x,y)_n=\left\{ \sigma\colon \Delta^n\times\Delta^1 \to \mathcal{C} \mid \sigma|_{\Delta^n\times\{0\}}\equiv x,\;\sigma|_{\Delta^n\times\{1\}}\equiv y \right\}$.

If $X$ is a complete Segal space, then we can associate to it a quasi-category $\mathcal{C}(X)$ via the rule $\mathcal{C}(X)_n:=X_{n,0}$, and this induces an equivalence between the $(\infty,1)$-category of complete Segal spaces and that of quasi-categories.

In particular, this functor should match up the hom spaces on both sides - we should have a homotopy equivalence ${\rm Map}_{\mathcal{C}(X)}(x,y)\simeq x\times^h_{X_0} X_1 \times^h_{X_0} y$ whenever $X$ is a complete Segal space and $x,y\in X_{0,0}$. Unfortunately, I'm a bit of a novice in higher category theory and don't know the literature well enough to know where to look for such a result.

So my question is the following: does anyone know where this equivalence of mapping spaces is explicitly proved (or possibly with ${\rm Map}$ replaced with any equivalent notion of mapping spaces for quasi-categories, such as ${\rm Map}^{\rm R}$ or ${\rm Map}^{\rm L}$)?