Warning 2.2.6.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories which are strictly isomorphic. Then $\operatorname{\mathcal{C}}$ is strict if and only if $\operatorname{\mathcal{D}}$ is strict. If we assume only that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are isomorphic (rather than strictly isomorphic), then we cannot draw the same conclusion. In other words, the condition that a $2$-category $\operatorname{\mathcal{C}}$ is strict is invariant under strict isomorphism, but not under isomorphism.
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