  We construct a wide subcategory of the category of finite association schemes
with a collection of desirable properties. Our subcategory has a first
isomorphism theorem analogous to that of groups. Also, standard constructions
taking schemes to groups (thin radicals and thin quotients) or algebras
(adjacency algebras) become functorial when restricted to our category. We use
our category to give a more conceptual account for a result of Hanaki
concerning products of characters of association schemes; i.e. we show that the
virtual representations of an association scheme form a module over the
representation ring of the thin quotient of the association scheme.
