  For a relational Horn theory $\mathbb{T}$, we provide useful sufficient
conditions for the exponentiability of objects and morphisms in the category
$\mathbb{T}\text{-}\mathsf{Mod}$ of $\mathbb{T}$-models; well-known examples of
such categories, which have found recent applications in the study of
programming language semantics, include the categories of preordered sets and
(extended) metric spaces. As a consequence, we obtain useful sufficient
conditions for $\mathbb{T}\text{-}\mathsf{Mod}$ to be cartesian closed, locally
cartesian closed, and even a quasitopos; in particular, we provide two
different explanations for the cartesian closure of the categories of
preordered and partially ordered sets. Our results recover (the sufficiency of)
certain conditions that have been shown by Niefield and Clementino--Hofmann to
characterize exponentiability in the category of partially ordered sets and the
category $V\text{-}\mathsf{Cat}$ of small $V$-categories for certain
commutative unital quantales $V$.
