We revisit the dichotomy between enriched and internal categories in a base category V. By describing these objects as monads internal to certain proarrow equipments, we construct a change-of-base adjunction between the category of enriched V-categories and the category of internal V-categories, provided V satisfies suitable conditions. This perspective allows us to cast enriched V-categories as internal V-categories whose object-of-objects is discrete, which finds applications in the study of the descent theory for V-functors, as shown in [3, Theorem 9.11].

Motivated by the study of the descent theory of functors between (T,V)-categories [1], the goal of [4] is to extend these techniques to the setting of generalized multicategories [2]. Using the above dichotomy as our guiding principle, we study the notion of change-of-base for a notion of lax algebras for monads in a suitable 2-category of proarrow equipments. Given suitable conditions on the category V and the monad T, we obtain an analogous adjunction between enriched and internal (T,V)-categories, which we use to describe effective descent functors of (T,V)-categories.