$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 6.3.2.4. Suppose we are given a diagram of simplicial sets
6.11
\begin{equation} \begin{gathered}\label{equation:localization-and-categorical-pushout-easy} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r]^{F_0} \ar [d]^-{T} & \operatorname{\mathcal{D}}_0 \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^{F} & \operatorname{\mathcal{D}}, } \end{gathered} \end{equation}
where $F_0$ exhibits $\operatorname{\mathcal{D}}_0$ as a localization of $\operatorname{\mathcal{C}}_0$ with respect to some collection of edges $W_0$. Then ( 6.11 ) is a categorical pushout square if and only if $F$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W = T(W_0)$.
Proof.
Let $\operatorname{\mathcal{E}}$ be an $\infty $-category. For any diagram $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, the composition $(G \circ F)|_{\operatorname{\mathcal{C}}_0}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{E}}$ factors through $F_0$, and therefore carries each edge of $W_0$ to an isomorphism in $\operatorname{\mathcal{E}}$. It follows that $G \circ F$ carries each edge of $W$ to an isomorphism in $\operatorname{\mathcal{E}}$. We therefore have a commutative diagram of $\infty $-categories
6.12
\begin{equation} \begin{gathered}\label{equation:localization-and-categorical-pushout-easy2} \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \ar [r]^{\circ F} \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d]^{ \circ T} \\ \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}) \ar [r]^{ \circ F_0} & \operatorname{Fun}( \operatorname{\mathcal{D}}_0[W_0^{-1}], \operatorname{\mathcal{E}}) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}). } \end{gathered} \end{equation}
The right side of (6.12) is a pullback square in which the horizontal maps are isofibrations (Remark 6.3.1.8), and therefore a categorical pullback square (Corollary 4.5.2.27). By assumption, precomposition with $F_0$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}_0, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}_0[W_0^{-1}], \operatorname{\mathcal{E}})$. Applying Propositions 4.5.2.18 and 4.5.2.21, we conclude that the outer rectangle is a categorical pullback square if and only if precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}[W^{-1}], \operatorname{\mathcal{E}})$. The desired result now follows by allowing the $\infty $-category $\operatorname{\mathcal{E}}$ to vary.
$\square$