# Definition:Dual Category

## Definition

Let $\mathbf C$ be a metacategory.

Its dual category, denoted $\mathbf C^{\text{op} }$, is defined as follows:

 Objects: $X^{\text{op} }$, for all $X \in \operatorname{ob}\mathbf C$ Morphisms: $f^{\text{op} }: D^{\text{op} } \to C^{\text{op} }$ for all $f: C \to D$ in $\mathbf C_1$ Composition: $\left({f^{\text{op} } \circ g^{\text{op} } }\right) := \left({g \circ f}\right)^{\text{op} }$, whenever this is defined Identity morphisms: $\operatorname{id}_{X^{\text{op} } } := \operatorname{id}_X^{\text{op} }$

It can be seen that this comes down to the metacategory obtained by reversing the direction of all morphisms of $\mathbf C$.

## Also known as

Many authors call $\mathbf C^{\text{op}}$ the opposite category of $\mathbf C$.

Others use e.g. $f^*$ in place of $f^{\text{op} }$. As the character $*$ is used so often already in mathematics, the form $f^{\text{op} }$ is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.