The concept of enriched adjunction is the generalization of that of adjoint functors (adjunctions in Cat) from category theory to enriched category theory.
(enriched adjunction)
For $\mathcal{V}$ a closed symmetric monoidal category with all limits and colimits, let $\mathcal{C}$, $\mathcal{D}$ be two $\mathcal{V}$-enriched categories. Then an adjoint pair of $\mathcal{V}$-enriched functors or enriched adjunction
is a pair of $\mathcal{V}$-enriched functors, as shown, such that there is a $\mathcal{V}$-enriched natural isomorphism between enriched hom-functors of the form
(e.g. Borceux 94, Def. 6.7.1)
The 2-functor $\mathcal{V}$-$Cat \rightarrow {Cat}$ that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction.
Max Kelly, section 1.11 of Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64, 1982, Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (pdf)
Francis Borceux, Vol 2, def. 6.2.4 of Handbook of Categorical Algebra, Cambridge University Press (1994)
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