Definition 1.9 (Equivalence of categories). Let $C$ and $D$ be categories. An equivalence between $C$ and $D$ consists of functors $F:C\to D$ and $G:D\to C$ together with natural isomorphisms $\alpha :1_C\to GF$, $\beta :FG\to 1_D$. We write $C\equiv D$ if there exists an equivalence between $C$ and $D$.

We say $P$ is a categorical property if