The category of categories gives rise to a notion of an isomorphism of categories, defined by interpreting Definition \ref{defn:iso} in $\textup{\textsf{cat}}$ or in $\textup{\textsf{cat}}$. Namely, an isomorphism of categories is given by a pair of inverse functors $F \colon \mathsf{C} \to \mathsf{D}$ and $G \colon \mathsf{D} \to \mathsf{C}$ so that the composites $GF$ and $FG$, respectively, equal the identity functors on $\mathsf{C}$ and on $\mathsf{D}$. An isomorphism induces a bijection between the objects of $\mathsf{C}$ and objects of $\mathsf{D}$ and likewise for the morphisms.