Let be a category. The dual category of is the category which has the same objects as , but in which all morphisms are “reversed”. That is to say if are objects of and we have a morphism , then we formally define an arrow in . is called the opposite arrow, or opposite morphism of . The composition is then defined to be . The dual category is sometimes called the opposite category and is denoted .

The category of Hopf algebras over a field is (equivalent to) the opposite category of affine group schemes over .

Categorical properties of lead directly to categorical properties of ; constructions on become constructions on . Usually such a construction is indicated with the prefix “co-”. For example, a coproduct is a product on the opposite category; this can be seen by looking at the commutative diagram that completely specifies a coproduct, and noting that it is the same as the diagram specifying a product with the arrows reversed. More generally, an inverse limit is a direct limit on the opposite category; for this reason, it is sometimes called a colimit. A cokernel is a kernel in the opposite category. Many other similar concepts exist.

If is a covariant functor from to some other category , then we can define, in a natural way, a contravariant functor from to , called the opposite functor of . In fact, this is often how contravariant functors are defined, and it is why most categorical theorems and constructions need not explicitly consider both cases.