Opposite category

Summary

The opposite category $\mathcal{C}^{\mathrm{op}}$ of a category $\mathcal{C}$ has the same objects and the same morphisms, with every arrow reversed. It turns monomorphisms into epimorphisms, products into coproducts, and limits into colimits. It is the basis of duality in category theory. It also lets one write a contravariant functor on $\mathcal{C}$ as an ordinary functor on $\mathcal{C}^{\mathrm{op}}$.