Relative symmetric monoidal closed categories I: Autoenrichment and change of base

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category $V$ its associated $V$-enriched category $\underline{V}$ extends to a 2-functor valued in an op-2-fibred 2-category of symmetric monoidal closed categories enriched over various bases. For a fixed $V$, we show that this induces a 2-functorial passage from symmetric monoidal closed categories $\textit{over}$ $V$ (i.e., equipped with a morphism to $V$) to symmetric monoidal closed $V$-categories over $\underline{V}$. As a consequence, we find that the enriched adjunction determined a symmetric monoidal closed adjunction can be obtained by applying a 2-functor and, consequently, is an adjunction in the 2-category of symmetric monoidal closed $V$-categories.

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