Joyal's CatLab Opposite categories

Definition

• The opposite of a category $C$ is the category $C^o$ obtained by formally reversing the direction of the arrows of $C$. It is sometime useful to make a distinction between the objects of $C$ and of $C^o$ by denoting $a^o\in C^o$ the object which corresponds to an object $a\in C$, and by denoting $f^o\in C^o(b^o,a^o)$ the arrow which corresponds to an arrow $f\in C(a,b)$. If $f\in C(a,b)$ and $g\in C(b,c)$ then the composite of $g^o\in C(c^o,b^o)$ with $f^o\in C(b^o,a^o)$ is defined to be the arrow $(g f)^o\in C^o(c^o,a^o)$. Thus $f^o g^o:=(g f)^o$.