When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify it in terms of its unit and counit. Both approaches (and their relationship) generalize to the enriched setting.
Given a functor and another , an adjunction is given by natural transformations and . These transformations must satisfy the equations and . By the weak Yoneda Lemma, this is equivalent to giving a -natural isomorphism .
Indeed, a -natural transformation in this direction must be of the form , and one in the other direction must be of the form . The equations and are equivalent, by the weak Yoneda Lemma, to the equations satisfied by the unit and counit of an adjunction.
The -functor that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction. The function underlying the -natural isomorphism is the bijection of this underlying adjunction.
As we saw before, a -functor has a left adjoint if and only if is representable for each . Also, an enriched equivalence is an enriched adjunction whose unit and counit are both -natural isomorphisms. Just as for ordinary adjunctions, we have transformations between enriched adjunctions, a category of enriched adjunctions between two enriched categories, enriched adjunctions with parameters, and so on.