full and faithful on mapping spaces

Question

I need to prove or find a counterexample to the following:

Let $C$ and $D$ be two $\infty$-categories. Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a full and faithful functor. Then for any $\infty$-category $\mathcal{A}$, the induces functor $\circ f: map(\mathcal{A}, \mathcal{C}) \rightarrow map(\mathcal{A}, \mathcal{D})$ is also full and faithful.

Here $map$ denotes the mapping space and $\circ f$ is just postcompose with $f$.

Answer 1

A proof of the statement can be found in Lax Colimits and free fibrations (Gepner-Haugseng-Nikolaus) Lemma 5.2.