# Antiautomorphism

I’ve been wanting to ask this for awhile.  What, precisely, is an “antiautomorphism?”  I think understanding that would help with problem 3.

An antiautomorphism of an algebra is the same as an automorphism of this algebra, except that instead of satisfying $f\left(ab\right) = f\left(a\right) f\left(b\right)$ (or equivalently $m\circ\left(f\otimes f\right) = f\circ m$) it must satisfy $f\left(ab\right) = f\left(b\right) f\left(a\right)$ (or equivalently $m\circ\left(f\otimes f\right) = f\circ m\circ\tau$, where $\tau$ is the twist/flip).
An antiautomorphism of a coalgebra is the same as an automorphism of this coalgebra, except that instead of satisfying $\left(f\otimes f\right)\circ\Delta = \Delta\circ f$ it must satisfy $\left(f\otimes f\right)\circ\Delta = \tau\circ \Delta\circ f$, where $\tau$ is the twist/flip.