Is the homotopy category of the quasicategory of quasifunctors between 1-categories $C,D$ equal to the 1-category of 1-functors $C\to D$?

Question

We use the ("geometric") model of $(\infty,1)$-categories as quasicategories.

Notation: For a 1-category $C$, let $\tilde{C}$ denote the $(\infty,1)$-category / quasicategory given by the simplicial nerve of $C$.

Let $C,D$ be two given 1-categories. Let $B$ be the $(\infty,1)$-category of $(\infty,1)$-functors $\tilde{C} \rightarrow \tilde{D}$, and let $A$ be the 1-category of 1-functors $C\rightarrow D$.

My question is, is $A$ naturally (isomorphic to) the homotopy 1-category $hB$ of $B$? Would anyone know of a reference where this is discussed?