In an $\infty$-topos, suppose we have two cartesian diagrams of the form $$ \require{AMScd} \begin{CD} \overline{A} @>>> \overline{B} \\ @VVV @VVV \\ A @>>> B . \end{CD} $$ Let $$ \begin{align} \varinjlim \left( \overline{A} \rightrightarrows \overline{B} \right) \stackrel{\sim}{\longrightarrow}\overline{C}, && \varinjlim \left( A \rightrightarrows B \right)\stackrel{\sim}{\longrightarrow} C \end{align} $$ be the resulting coequalizers. Is the diagram $$\require{amscd}\begin{CD} \overline{B} @>>> \overline{C} \\ @VVV @VVV \\ B @>>> C \end{CD}$$ cartesian?

I think this is a formal consequence of Mather's First Cube Theorem (or Magic Cube Lemma), but I also don't know if I can use this theorem in any $\infty$-topos.