Lemma 4.44.2. Let $\mathcal{C}$ be a $(2,1)$-category. Assume given a $2$-commutative diagram

in $\mathcal{C}$, where the right square is $2$-cartesian with respect to a $2$-isomorphism $\phi \colon g \circ p \to f \circ q$. Choose a $2$-arrow $\gamma ' : y' \circ j \to p \circ x'$. Set $x = q \circ x'$, $y = g \circ y'$ and let $\gamma : y \circ j \to f \circ x$ be the $2$-isomorphism $\gamma = (\phi \star \text{id}_{x'}) \circ (\text{id}_ g \star \gamma ')$. Then the category $\mathcal{D}'$ of dotted arrows for the left square and $\gamma '$ is equivalent to the category $\mathcal{D}$ of dotted arrows for the outer rectangle and $\gamma $.

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