Opposite category

The opposite category $$\mathsf{C}^\text{op}$$ of a category $$\mathsf{C}$$ is the category with


 * $$\operatorname{Ob} \mathsf{C}^\text{op} := \operatorname{Ob} \mathsf{C}$$,
 * $$\operatorname{Hom}_{\mathsf{C}^\text{op}}(X, Y) := \operatorname{Hom}_{\mathsf{C}^\text{op}}(Y, X)$$, and
 * $$f \circ_{\mathsf{C}^\text{op}} g := g \circ_{\mathsf{C}} f$$.

It is immediate to check to that this forms a category.

Uses

 * The main use for opposite categories is in defining contravariant functors.