  Cofunctors are a kind of map between categories which lift morphisms along an
object assignment. In this paper, we introduce cofunctors between categories
enriched in a distributive monoidal category. We define a double category of
enriched categories, enriched functors, and enriched cofunctors, whose
horizontal and vertical 2-categories have 2-cells given by enriched natural
transformations between functors and cofunctors, respectively. Enriched lenses
are defined as a compatible enriched functor and enriched cofunctor pair;
weighted lenses, which were introduced by Perrone, are precisely lenses
enriched in weighted sets. Several other examples are also studied in detail.
