Warning 2.2.8.19. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category. We can then consider two different notions of isomorphism in $\operatorname{\mathcal{C}}$:
We say that a morphism $f: X \rightarrow Y$ is a strict isomorphism if it is an isomorphism in the underlying category of $\operatorname{\mathcal{C}}$: that is, if there exists a $1$-morphism $g: Y \rightarrow X$ satisfying $g \circ f = \operatorname{id}_ X$ and $f \circ g = \operatorname{id}_ Y$.
We say that a morphism $f: X \rightarrow Y$ is an isomorphism if the homotopy class $[f]$ is an isomorphism in the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$: that is, if there exists a $1$-morphism $g: Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are isomorphic to $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ as objects of the categories $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ and $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Y)$, respectively.
Every strict isomorphism in $\operatorname{\mathcal{C}}$ is an isomorphism. However, the converse is false in general (see Example 2.2.8.20).