$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. Then the induced functors

\[ F_{q/}: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ q)/} \quad \quad F_{/q}: \operatorname{\mathcal{C}}_{/q} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ q)} \]

are also conservative.

Proof. We will show that the functor $F_{/q}$ is conservative; the conservativity of $F_{q/}$ follows by a similar argument. Let $\pi : \operatorname{\mathcal{C}}_{/q} \rightarrow \operatorname{\mathcal{C}}$ and $\pi ': \operatorname{\mathcal{D}}_{/(F \circ q)} \rightarrow \operatorname{\mathcal{D}}$ denote the projection maps. Then $\pi $ and $\pi '$ are right fibrations of $\infty $-categories (Proposition, and therefore conservative (Proposition Since $F$ is conservative, from Remark that the functor $F \circ \pi = \pi ' \circ F_{/q}$ is also conservative. Applying Remark again, we conclude that $F_{/q}$ is conservative. $\square$