Enriched Category Basics

The notion of an enriched category generalizes the concept of category by replacing the hom-sets of an ordinary category by objects of an arbitrary monoidal category. In this article we give a formal definition of enriched categories and we give formal proofs of a relatively narrow selection of facts about them. One of the main results is a proof that a closed monoidal category can be regarded as a category "enriched in itself". The other main result is a proof of a version of the Yoneda Lemma for enriched categories.