something else in [@S80], in [@S04] one finds *autonomous category* and *Frobenius pseudomonoid*, and in [@L06] *ambidextrous adjunction*.)

A *self-adjunction* is an adjunction where the categories $\cal A$ and $\cal B$ are equal, and the functors $F$ and $G$, which are now endofunctors, are also equal. We write in this context $\cal S$ for $\cal A$ and $\cal B$, and $F$ for both $F$ and $G$. So the unit and counit of a self-adjunction are respectively ${\gamma\!:I_{\cal S}\strt FF}$ and ${\varphi\!:FF\strt I_{\cal S}}$. Every self-adjunction is a bijunction. (Self-adjunction is not often mentioned in textbooks of category theory—an exception is [@AHS], Chapter 5, Exercise 19G; the notion of self-adjoint functor of [@T99] is a related but different notion.)

The bijunction freely generated by a single object $0$ on the $\cal A$ side is defined in syntactical terms analogously to *Frob* in Section 2. The objects of the category $\cal A$ are here $0$, $PU0$, $PUPU0$, etc., while those of $\cal B$ are $U0$, $UPU0$, $UPUPU0$, etc.

We define analogously the free self-adjunction generated by a single object $0$. An object of the category $\cal S$ of this self-adjunction is of the form $F^n0$, where $F^n$ is a sequence of ${n\geq 