# Opposite category

In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, $(C^{op})^{op} = C$.

## Examples

• An example comes from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤new by
xnew y if and only if yx.
For example, there are opposite pairs child/parent, or descendant/ancestor.

## Properties

Opposite preserves products:

$(C\times D)^{op} \cong C^{op}\times D^{op}$ (see product category)

Opposite preserves functors:

$(\mathrm{Funct}(C,D))^{op} \cong \mathrm{Funct}(C^{op},D^{op})$[2][3] (see functor category, opposite functor)

Opposite preserves slices:

$(F\downarrow G)^{op} \cong (G^{op}\downarrow F^{op})$ (see comma category)