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Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon\mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and acartesian monad $ T $ on an extensive category $ \mathcal V $ with finitelimits, we construct an adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcalV(1,-) \colon \mathsf{Cat}(T,\mathcal V) \to (\overline T, \mathcalV)\text{-}\mathsf{Cat} $ between categories of generalized enrichedmulticategories and generalized internal multicategories, provided the monad $T $ satisfies a suitable condition, which is satisfied by several examples.  We verify, moreover, the left adjoint is fully faithful, and preservespullbacks, provided that the copower functor $ - \boldsymbol{\cdot} 1 \colon\mathsf{Set} \to \mathcal V $ is fully faithful. We also apply this result tostudy descent theory of generalized enriched multicategorical structures.  These results are built upon the study of base-change for generalizedmulticategories, which, in turn, was carried out in the context of categoriesof horizontal lax algebras arising out of a monad in a suitable 2-category ofpseudodouble categories.

Generalized multicategories: change-of-base, embedding, and descent