Frobenius equations are replaced by Lawvere’s equations).

The category *Frob* has a strict monoidal structure. The $\otimes$ of this monoidal structure is addition on objects. We define ${\mj_n\otimes f}$ as $M^n f$, where $M^n$ is a sequence of ${n\geq 0}$ occurrences of $M$, while ${f\otimes\mj_n}$ is defined by increasing the subscripts of $f$ by the natural number $n$. Then for ${f_1\!:n_1\str m_1}$ and ${f_2\!:n_2\str m_2}$ we have $$f_1\otimes f_2=_{df}(f_1\otimes\mj_{m_2})\cirk(\mj_{n_1}\otimes
f_2).$$ The category *Frob* was envisaged as a monoidal category in [@K03] (Section 3.6.16).

The category $\cal  M$ of the monad freely generated by a single object $0$ is defined like *Frob* save that we omit the arrow terms $\varepsilon^\Box_n$ and $\delta^\Box_n$, and whatever involves them. By omitting $\varepsilon^\Diamond_n$ and $\delta^\Diamond_n$, we define analogously the comonad freely generated by $0$.

Free adjunctions and monads
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An adjunction is given by two categories $\cal  A$ and $\cal  B$, a functor $F$ from $\cal  B$ to $\cal  A$, the *left adjoint*, a functor $G$ from $\cal  A$ to $\cal  B$, the *right adjoint*, a natural transformation ${\gamma\!:I_{\cal
B}\strt GF}$, the *unit* of the adjunction, and a natural transformation ${\varphi\!:FG\strt I_{\cal A}}$, the *counit* of the adjunction, which satisfy the following *triangular* equations for 