  Using generalized enriched categories, in this paper we show that
Rosick\'{y}'s proof of cartesian closedness of the exact completion of the
category of topological spaces can be extended to a wide range of topological
categories over $\mathsf{Set}$, like metric spaces, approach spaces,
ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In
order to do so we prove a sufficient criterion for exponentiability of
$(\mathbb{T},V)$-categories and show that, under suitable conditions, every
$(\mathbb{T},V)$-injective category is exponentiable in
$(\mathbb{T},V)\text{-}\mathsf{Cat}$.
