$\quad$
- $\textup{\textsf{cat}}_R^\mathrm{op}$ is the category whose objects are non-zero natural numbers and in which a morphism from $m$ to $n$ is an $m \times n$ matrix with values in $R$. The upshot is that a reader who would have preferred the opposite handedness conventions when defining $\textup{\textsf{cat}}_R$ would have lost nothing by adopting them.
- When a preorder $(\mathsf{P},\leq)$ is regarded as a category, its opposite category is the category that has a morphism $x \to y$ if and only if $y \leq x$. For example, $\bbomega^\mathrm{op}$ is the category freely generated by the graph
\(\cdots \to 3 \to 2 \to 1 \to 0.\)
- If $G$ is a group, regarded as a one-object groupoid, the category $(\mathsf{B} G)^\mathrm{op} \cong \mathsf{B} (G^\mathrm{op})$ is again a one-object groupoid, and hence a group. The group $G^\mathrm{op}$ is called the opposite group and is used to define right actions as a special case of left actions; see Example \ref{ex:G-actions}.