Definition:Dual Category

Definition
Let $\mathbf C$ be a metacategory.

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

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.

Also see

 * Dual Category is Category