On bi-enriched -categories

We extend Lurie's definition of enriched -categories to notions of left enriched, right enriched and bi-enriched -categories, which generalize the concepts of closed left tensored, right tensored and bitensored -categories and share many desirable features with them. We use bi-enriched -categories to endow the -category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched -categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we prove an end formula for morphism objects of enriched -categories of enriched functors and compute the monad for enriched functors. We build our theory closely related to Lurie's higher algebra: we construct an enriched -category of enriched presheaves via the enveloping tensored -category, construct transfer of enrichment via scalar extension of bitensored -categories, and construct enriched Kan-extensions via operadic Kan extensions. In particular, we develop an independent theory of enriched -categories for Lurie's model of enriched -categories.