Construction 2.2.8.27 (The Core of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{\mathcal{C}}^{\simeq }$ as follows:
The objects of $\operatorname{\mathcal{C}}^{\simeq }$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}^{\simeq } }( X, Y)$ is the full subcategory of $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ spanned by the isomorphisms $f: X \rightarrow Y$.
The composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}^{\simeq }$ are obtained by restricting the composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}$ (which is well-defined by virtue of Remark 2.2.8.22).
We will refer to $\operatorname{\mathcal{C}}^{\simeq }$ as the core of the $2$-category $\operatorname{\mathcal{C}}$.