  We prove that the category of models of any relational Horn theory satisfying
a mild syntactic condition is infinitely extensive. Central examples of such
categories include the categories of preordered sets and partially ordered
sets, and the categories of small $\mathscr{V}$-categories, (symmetric)
pseudo-$\mathscr{V}$-metric spaces, and (symmetric) $\mathscr{V}$-metric spaces
for a commutative unital quantale $\mathscr{V}$. We also explicitly
characterize initial sources and final sinks in such categories, and in
particular embeddings and quotients.
