algebra (see [@K03] and references therein), while $\varepsilon_0$ is usually deemed interesting only for set-theorists and proof-theorists.

The categorial abstraction of the notion of Frobenius algebra leads to the notion of Frobenius monad (for some more details, see below). The structure of a Frobenius monad is given by a category with an endofunctor that bears both the structure of a monad (or triple) and a comonad, and satisfies moreover additional conditions called Frobenius equations (see the next section).

The notion of Frobenius monad is closely related to a special kind of adjoint situation where two functors (not necessarily distinct) are both left and right adjoint to each other (see [@M65], [@CMS97], [@K02], [@M03], [@S04], [@L06], [@CH09], and further references in these papers). Adjunction is a central notion in category theory, in logic, and perhaps in mathematics in general (see [@ML98] and [@LAW69a]), and the connection of this notion with the notion of Frobenius monad may serve to explain the importance of the latter.

One of the goals of this paper is to show that the notion of adjunction where two functors are both left and right adjoint to each other amounts, in a sense that we will make precise, to 