every object $B$ of $\cal  B$ and every object $A$ of $\cal  A$: $$\varphi_{FB}\cirk
F\gamma_B=\mj_{FB},\hspace{5em}G\varphi_A\cirk\gamma_{GA}=\mj_{GA}.$$

The adjunction freely generated by a single object $0$ on the $\cal  B$ side is defined in syntactical terms analogously to *Frob* (see [@D99], Chapter 4, for a detailed exposition). In this free adjunction, the objects of $\cal  B$ are $0$, $GF0$, $GFGF0$, etc., while those of $\cal  A$ are $F0$, $FGF0$, $FGFGF0$, etc. This notion of freely generated adjunction is essentially the same as a 2-categorial notion that may be found in [@A74], [@SS86] (cf. also [@KS74]) and [@L08]. If we consider the sub-2-category of the 2-category *Cat* of categories whose only 0-cells are $\cal  A$ and $\cal  B$, whose 1-cells are made of $F$ and $G$, and whose 2-cells are made of $\varphi$, $\gamma$, $F$ and $G$, we obtain a 2-category isomorphic in the 2-categorial sense to the free category *Ad* of [@A74] (called *Adj* in [@SS86]). This does not depend on the number of generators of our free adjunction, provided it is not zero, and they may be either on the $\cal  A$ side or on the $\cal  B$ side.

The connection of our notion of free adjunction with the 2-category *Ad* 