Opposite category

Remarks

The category axioms for $\mathcal{C}^{\mathrm{op}}$ follow from those of $\mathcal{C}$ read backwards, and $(\mathcal{C}^{\mathrm{op}})^{\mathrm{op}} = \mathcal{C}$. Example: let $(P, \le)$ be a preorder, seen as a category with one morphism $a \to b$ when $a \le b$. Its opposite has one morphism $a \to b$ when $b \le a$, so it is the preorder $(P, \ge)$. Non-example: the rule "keep the arrows as they are and only reverse the order of composition" does not give a category in general. For $f \colon A \to B$ and $g \colon B \to C$, the expression $f \circ g$ has no meaning in $\mathcal{C}$ unless $C = A$. A morphism $f$ is a monomorphism in $\mathcal{C}$ exactly when $f^{\mathrm{op}}$ is an epimorphism in $\mathcal{C}^{\mathrm{op}}$.