  This is half an overview article since what we describe here is essentially
known. We describe $KK$-theory by generators and relations in a formal sum of
formal products of $*$-homomorphisms and some synthetical morphisms. What comes
out is a category. The Kasparov product is then just the composition of
morphisms. This description may be interesting to anyone who wants a quick and
elementary definition of $KK$-theory. The description could also be used for
other categories of algebras than $C^*$-algebras endowed with group actions,
for example, $C^*$-algebras equipped with an action by a semigroup, a category
et cetera.
