We introduce a category of cluster algebras with fixed initial seeds. This
category has countable coproducts, which can be constructed combinatorially,
but no products. We characterise isomorphisms and monomorphisms in this
category and provide combinatorial methods for constructing special classes of
monomorphisms and epimorphisms. In the case of cluster algebras from surfaces,
we describe interactions between this category and the geometry of the
surfaces.Comment: 37 page