Warning 2.2.1.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category. If $\operatorname{\mathcal{C}}$ is strict, then we can extract from $\operatorname{\mathcal{C}}$ an underlying ordinary category having the same objects and $1$-morphisms (Remark 2.2.0.3). However, this operation has no counterpart for a general $2$-category $\operatorname{\mathcal{C}}$: in general, composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ is associative only up to isomorphism.
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