  Enrichment and internal categories are two different way to generalize the
notion of category. As such, enriching double categories (which are categories
internal to Cat) is not a clear concepts. One can look at the internal
categories of the category of C-enriched categories or the internal C-category
in Cat. This concepts, unfortunately, are not powerful enough, even thin
(posetal) double categories can't be seen this way. However, an orthogonal
investigation, that of the enrichment by a double category, will help relaxing
the internal C-category to a incorporate the internal categories of the
category of C-enriched categories, as well as thin double categories or framed
bicategories.
