Remark 2.2.8.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $2$-categories. Then $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$ (see Remark 2.2.8.16). Beware that the analogous assertion need not hold if we assume only that $F$ is a lax functor of $2$-categories.
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