Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$.
Its well known that there are (faithful and locally full and faithful) immersions $(-)_\bullet: Cat \to Prof ,\ (-)^\bullet: Cat^{op} \to Prof $ (where $Cat$ is the usual 2-category, and $Cat^{op}$ dualizes (reverses) arrows only).
We can extend the (cartesian) monoidal products:
defining for $ \mathscr{A} \dashrightarrow B$ and $E: \mathscr{C} \dashrightarrow \mathscr{D}$ the profunctor $D \times E: \mathscr{A} \times \mathscr{C} \dashrightarrow \mathscr{B} \times \mathscr{D}$ as the composition $(\mathscr{A} \times \mathscr{C})^{op} \times (\mathscr{B} \times \mathscr{D}) \cong (\mathscr{A} \times \mathscr{B}^{op} )\times (\mathscr{C} \times \mathscr{D}^{op}) \xrightarrow{D\times E } Set \times Set \xrightarrow{\times } Set$
If I'm not wrong the immersions $(-)_\bullet,\ (-)^\bullet$ come out strict monoidal.
I ask if $Prof$ has also a (monoidal) closed structure (I tried in vain to get it..)
Edit: changed $\otimes$ in $\times$ (At first I thought the problem into enriched context).