To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is the bicategory $Prof(\mathscr{C})$ of internal categories of $\mathscr{C}$ and profunctors. (see for example first pages of "Fibrations and Yoneda's lemma in a 2-category" by R.Street and the definitions of internal profunctor in 'Topos theory' of P. Johnstone 1978).

Now given a morphism $f: X \to Y$ in $\mathscr{C}$ and a monoid $B \rightrightarrows Y$ (we call it also a internal category on $Y$) the pullback by $f$ give a monoid $f^*B$ on $X$, furthermore given a internal functors $(f_1, f): (A \rightrightarrows X) \to (B \rightrightarrows Y)$, it correspond equivalently to a morphism of monoids (i.e. a internal functor) $A \to f^*B$ on $X$. Then the category of internal funtors in $\mathscr{C}$ is equivalent to a fibred category $\mathscr{F}$ on $\mathscr{C}$ with fibres $\mathscr{F}(X)=$''*internal categories (and functors) on $X$*''.

From 'Bicategories of spans and relations' A. Carboni, S. Kasangian (JPAA 33, 1984) the bicategory $Span(\mathscr{C})$ and the category $\mathscr{C}$ are strictly related ( I seems that $\mathscr{C}$ is 2-equivalent (as locally discrete 2-category) to the maps (morphisms that are right adjoint) of $Span(\mathscr{C})$).

Now, considering internal profunctors (horizontally) and internal functors (vertically) we get a **pseudo-double-category** (for definition see for example http://arxiv.org/abs/math/0604549). The some if $\mathscr{C}$ has one object (i.e. is a monoidal category) and consider the (bi)modules (horizontally) and morphisms of monoids (vertically).

1) I ask:

If the usual construction of the bicategory of modules (from a bicategory $\mathscr{S}$ ) is generalizable to a construction of a pseudo-double-category of modules as horizontal arrows, and monoids morphisms (of a different type from the modules) as vertical arrows, this from a bicategory $\mathscr{S}$ and some fibration on some category $\mathscr{C}$ related to $\mathscr{S}$ (may be $\mathscr{S}$ more general than a spans bicategory).

And if (possibly) we can get decomposition theorems similar to that of Benabou about the decomposition of a profunctor (see 'Topos Theory' P. Johnstone Th.2.48 page 63.)

2) Given a monoidal category $\mathbf{V}$ (symmetrical, closed if you want).

I ask if:

Is knowed in mathematical literature a construction of a bicategory $\mathscr{S}$ such that its bicategory of bimodule $Mod(\mathscr{S})$ is (equivalent to) the bicategory of $\mathbf{V}$-profuntors on (small) $\mathbf{V}$-categories ?

notthink of profunctors as of (discrete) fibrations (because this view is not even true for enriched categories). Instead you may think of profunctors as of (codiscrete) cofibrations. You should definitely take a look into the paper mentioned in the above. $\endgroup$2more comments