  It is a classical theorem that the free product of ordered groups is
orderable. In this note we show that, using a method of G. Bergman, an ordering
of the free product can be constructed in a functorial manner, in the category
of ordered groups and order-preserving homomorphisms. With this functor
interpreted as a tensor product this category becomes a tensor (or monoidal)
category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$
with the natural topology, then for fixed groups $F$ and $G$ our construction
can be considered a function $O(F) \times O(G) \to O(F * G)$. We show that this
function is continuous and injective. Similar results hold for left-ordered
groups.
