Cartesian closed hull of the category of uniform spaces

A concrete category $\text{K}$ is a CCT (cartesian closed topological) extension of the category Unif of uniform spaces if 1. $\text{K}$ is cartesian closed, 2. Unif is a full, finitely productive subcategory of $\text{K}$ and the forgetful functor of $\text{K}$ extends that of Unif and 3. $\text{K}$ has initial structures. We describe the smallest CCT extension of Unif which is called the CCT hull by H. Herrlich and L.D. Nel. The objects of the CCT hull are bornological uniform spaces, i.e. uniform spaces endowed with a collection of “bounded” sets related naturally to the uniformity; the morphisms are the uniformly continuous maps which preserve the bounded sets.