\begin{definition}[Definition:Dual Category]
Let $\mathbf C$ be a metacategory.
Its '''dual category''', denoted $\mathbf C^{\text{op} }$, is defined as follows:
{{DefineCategory
|ob = $X^{\text{op} }$, for all $X \in \operatorname{ob}\mathbf C$
|mor = $f^{\text{op} }: D^{\text{op} } \to C^{\text{op} }$ for all $f: C \to D$ in $\mathbf C_1$
|comp = $\left({f^{\text{op} } \circ g^{\text{op} } }\right) := \left({g \circ f}\right)^{\text{op} }$, whenever this is defined
|id = $\operatorname{id}_{X^{\text{op} } } := \operatorname{id}_X^{\text{op} }$
}}
It can be seen that this comes down to the metacategory obtained by reversing the direction of all morphisms of $\mathbf C$.
\end{definition}