Remark 4.6.2.7. Suppose we are given a categorical pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{q} & \operatorname{\mathcal{D}}\ar [d] \\ \operatorname{\mathcal{C}}' \ar [r]^-{F'} & \operatorname{\mathcal{D}}'. } \]
If $F'$ is fully faithful, then $F$ is fully faithful (see Remark 4.6.2.6 and Corollary 3.4.1.5).