I am interested in the following property that a bicategory may or may not have.

Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\rightarrow \mathbf{B}(x,z)$ for an arbitrary object $z$ via precomposition $g\mapsto g\ast f$. Similarly, there is a postcomposition functor.

We claim that

($\ast$) The pre- and postcompositions with any $1$-morphism $f$ (with respect to an arbitrary object $z$) both have right adjoints $f_{!}$ and $f^{!}$.

**Examples:** 1. A bicategory with one object aka monoidal category has property ($\ast$) if and only if it is closed (sometimes called biclosed). Thus the title of the question.

The bicategory with rings as objects, $(R,S)$-bimodules as $1$-morphisms (composition: tensor product) and $(R,S)$-linear maps as $2$-morphisms also has this property. If $M$ is an $(R,S)$-bimodule and $N$ is an $(R,T)$-bimodule, then $M_{!}(N)$ is given by $\mathrm{Hom}_R(M,N)$ which is an $(S,T)$-bimodule.

The bicategory of profunctors $\mathrm{Prof}(\mathcal{V})$ where $\mathcal{V}$ is a cosmos (similar construction to 2).

**Question:** Has this notion been studied?