Given a bicomplete closed symmetric monoidal category $\mathcal V$, denote by $\operatorname{Cat}(\mathcal V)$ the category of small $\mathcal V$-enriched categories. The object set functor $$\operatorname{Ob}\colon\operatorname{Cat}(\mathcal V)\longrightarrow\operatorname{Set}$$ is a Grothendieck bifibration.

Question 1: Does this bifibration satisfy the Beck-Chevalley condition?

A lax symmetric monoidal functor $G\colon \mathcal W\rightarrow \mathcal V$ between such monoidal categories induces a cartesian functor $$G\colon \operatorname{Cat}(\mathcal W)\longrightarrow \operatorname{Cat}(\mathcal V)$$ in the obvious way.

If it fits into an adjoint pair $F\colon\mathcal V\rightleftarrows\mathcal W\colon G$ then $F$ is colax monoidal, so it doesn't give rise to a functor between categories of enriched categories in a straightforward way as above. Nevertheless, there is an adjoint pair $$F^{cat}\colon \operatorname{Cat}(\mathcal V)\rightleftarrows \operatorname{Cat}(\mathcal W)\colon G$$ where $F^{cat}$ need not be defined by $F$ on morphisms.

Question 2: Is $F^{cat}$ a fibered left adjoint?

I mean in the sense of Definition 8.4.1 of Borceux's 2nd book.

If any of the answers are positive under some extra, not very restrictive assumptions, I would also be very glad to know. I'm mostly interested in the second question. It would also be great if you could point me to references where this is considered.

I should remark that $\operatorname{Cat}(\mathcal V)$ is the category of algebras over a monad in the category $\operatorname{Graph}(\mathcal V)$ of enriched graphs. This category fits into a similar bifibration and the monad is compatible with it. If we replace $\operatorname{Cat}(\mathcal V)$ with $\operatorname{Graph}(\mathcal V)$ in the questions above, the answers are positive for obvious reasons. However, this doesn't seem to help at all.


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