equations hold: $$M\delta^\Diamond_A\cirk\delta^\Box_{MA}=\delta^\Diamond_{MA}\cirk
M\delta^\Box_A=\delta^\Box_A\cirk\delta^\Diamond_A.$$ (For easier comparison, we use here, with slight modifications, the notation with the modal superscripts $\Box$ and $\Diamond$, which was introduced in [@DP08b].)

The equations defining the notions of monad and comonad are given below. For the Frobenius equations the reader may consult [@K03] (in particular, Lemma 2.3.19, and [@DP08b], Sections 6-7; as far as we know, and according to [@K06], the first appearance of these equations is in [@CW87]). Lawvere introduced in [@LAW69] (pp. 151-152) the notion of Frobenius monad with the equations $$M\varepsilon^\Box_A\cirk M\delta^\Diamond_A\cirk\delta^\Box_{MA}=
\varepsilon^\Box_{MA}\cirk\delta^\Diamond_{MA}\cirk
M\delta^\Box_A=\delta^\Diamond_A,$$ or, alternatively, the dual equations $$\delta^\Diamond_{MA}\cirk M\delta^\Box_A\cirk
M\varepsilon^\Diamond_A=
M\delta^\Diamond_A\cirk\delta^\Box_{MA}\cirk\varepsilon^\Diamond_{MA}=
\delta^\Box_A,$$ which can replace the Frobenius equations. (In the terminology of [@DP08b], Section 8, a Frobenius monad is a dyad, or codyad, where $\Box$ and $\Diamond$ coincide.)

The category *Frob* of the Frobenius monad freely generated by a single object, denoted by $0$, has as objects the natural numbers ${n\geq 0}$, where $n$ stands for a sequence of $n$ occurrences of $M$; so $Mn$ is ${n\pl 1}$. The arrows of this category are defined syntactically as equivalence classes of *arrow terms*, which are defined inductively as follows. The primitive arrow terms of *Frob* are

$\mj_n\!:n\str n$,\
=$\varepsilon^\Box_n\,$=$:n\pl 1\str
n$,= 