Frobenius monad in the last section of the paper. We leave however as an open question what is the exact categorial abstraction of the notion of Frobenius algebra.

This paper is a companion to [@DP08b], but, except for some side comments, an acquaintance with that paper is not indispensable. We rely however, as we said above, on the results of [@DP03], and we assume an acquaintance with parts of that earlier paper, though some of the essential matters we need are reviewed in Section 6. We assume also the reader is acquainted with some basic notions of category theory, which may all be found in [@ML98], but, for the sake of notation, we define some of these basic notions below.

The free Frobenius monad
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A *Frobenius monad* is a structure made of a category $\cal
A$, an endofunctor $M$ of $\cal  A$ (i.e. a functor from $\cal A$ to $\cal  A$) and the natural transformations

=$\varepsilon^\Box\,$=$:M\strt I_{\cal
A}$,= $\varepsilon^\Diamond\,$=$:I_{\cal A}\strt M$,\
$\delta^\Box$$:M\strt MM$, $\delta^\Diamond$$:MM\strt M$,

for $I_{\cal A}$ being the identity functor of $\cal  A$, such that $\langle{\cal
A},M,\varepsilon^\Diamond,\delta^\Diamond\rangle$ is a monad, $\langle{\cal A},M,\varepsilon^\Box,\delta^\Box\rangle$ is a comonad, and, moreover, for every object $A$ of $\cal  A$, the following *Frobenius* 