$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 4.8.5.29. Let $n \geq 0$ be an integer, and suppose we are given a categorical pullback diagram of $\infty $-categories
4.83
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-locally-truncated} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [d]^{F'} \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r]^-{G} & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}
If $F$ is $n$-full, then $F'$ is $n$-full. The converse holds if $G$ is full and essentially surjective.
Proof.
The case $n = 0$ follows from Remark 4.6.2.19. We will therefore assume that $n \geq 1$. Using Corollary 4.5.2.23, we can reduce to the case where $F$ and $G$ are isofibrations. In this case, our assumption that (4.83) is a categorical pullback square guarantees that the induced map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an equivalence of $\infty $-categories (Proposition 4.5.2.26). Using Remark 4.8.5.18, $\operatorname{\mathcal{C}}'$ by the fiber product $\operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ and thereby reduce to the case where the diagram (4.83) is a pullback square. Note that, if the functor $G$ is full and essentially surjective, then every morphism of $\operatorname{\mathcal{D}}$ can be lifted to a morphism of $\operatorname{\mathcal{D}}'$. The desired result now follows from the criterion of Proposition 4.8.5.27.
$\square$