Let $\mathcal{C}$ be a category. The dual category $\mathcal{C}^{*}$ of $\mathcal{C}$ is the category which has the same objects as $\mathcal{C}$ , but in which all morphisms are ``reversed''. That is to say if $A,B$ are objects of $\mathcal{C}$ and we have a morphism $f: A \to B$ , then we formally define an arrow $f^{*}: B \to A$ in $\mathcal{C}^{*}$ . $f^*$ is called the opposite arrow, or opposite morphism of $f$ . The composition $f^{*}\circ g^{*}$ is then defined to be $(g\circ f)^{*}$ . The dual category is sometimes called the opposite category and is denoted $\mathcal{C}^{\op}$ .

The category of Hopf algebras over a field $k$ is (equivalent to) the opposite category of affine group schemes over $\operatorname{spec} k$ .

Categorical properties of $\mathcal{C}$ lead directly to categorical properties of $\mathcal{C}^{\op}$ ; constructions on $\mathcal{C}$ become constructions on $\mathcal{C}^{\op}$ . 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 $F$ is a covariant functor from $\mathcal{C}$ to some other category $\mathcal{D}$ , then we can define, in a natural way, a contravariant functor $F^{\op}$ from $C^{\op}$ to $D$ , called the opposite functor of $F$ . 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.