$\varepsilon^\Diamond_n\,$=$:n\str n\pl 1$,\
$\delta^\Box_n$$:n\pl 1\str n\pl 2$, $\delta^\Diamond_n$$:n\pl 2\str n\pl 1$.

The remaining arrow terms of *Frob* are defined inductively out of these with the clauses:

īf ${f\!:n\str m}$ and ${g\!:m\str k}$ are arrow terms, then so is ${(g\cirk f)\!:n\str k}$;\
if ${f\!:n\str m}$ is an arrow term, then so is ${Mf\!:n\pl
1\str m\pl 1}$.

We take for granted the outermost parentheses of arrow terms, and omit them. (Further omissions of parentheses will be permitted by the associativity of $\cirk$.)

The least equivalence relation, congruent with respect to $\cirk$ and $M$, by which we obtain the arrows of *Frob* is such that, first, we have the *categorial* equations of composition with $\mj$ and associativity of composition $\cirk$, and the *functorial* equations for $M$ (see [@DP08b], Section 2). We have next the *naturality* equations:

($\varepsilon^\Box$ [*nat*]{})=$f\cirk\varepsilon^\Box_n=\varepsilon^\Box_m\cirk
Mf$,($\varepsilon^\Diamond$ [*nat*]{})=$\varepsilon^\Diamond_m\cirk
f=Mf\cirk\varepsilon^\Diamond_n$,\
($\delta^\Box$ [*nat*]{})$MMf\cirk\delta^\Box_n=\delta^\Box_m\cirk
Mf$,($\delta^\Diamond$ [*nat*]{})$\delta^\Diamond_m\cirk
MMf=Mf\cirk\delta^\Diamond_n$,

the *comonad* and *monad* equations:

=$(\delta^\Box)$=$M\delta^\Box_n\!$=$\cirk\delta^\Box_n=
\delta^\Box_{n+1}\cirk\delta^\Box_n$,= $(\delta^\Diamond)$=$\delta^\Diamond_n\cirk
M\delta^\Diamond_n$=$=\delta^\Diamond_n\cirk
\delta^\Diamond_{n+1}$,\
$(\Box\beta)$ $\varepsilon^\Box_{n+1}$$\cirk\delta^\Box_n=\mj_{n+1}$, $(\Diamond\beta)$$\delta^\Diamond_n\cirk
\varepsilon^\Diamond_{n+1}$$=\mj_{n+1}$,\
$(\Box\eta)$ $M\varepsilon^\Box_n$$\cirk\delta^\Box_n=\mj_{n+1}$, $(\Diamond\eta)$$\delta^\Diamond_n\cirk
M\varepsilon^\Diamond_n$$=\mj_{n+1}$,

and, finally, the *Frobenius* equations where $A$ is replaced by $n$. The equations $(\delta^\Box)$ and $(\delta^\Diamond)$ are redundant in this axiomatization (see [@K03], Proposition 2.3.24, and [@DP08b], Section 6; they do not seem however to be redundant when the 