I'm reading Radford's text on Hopf Algebras right now and I'm a bit confused about the definition of a left A-Hopf algebra. The definition given in the book is:

Let $A$ be a $\Bbbk$-bialgebra. A $\underline{\text{left $A$-Hopf module}}$ is a triple ($M$, $\mu$, $\rho$) where ($M, \mu$) is a left $A$-module, ($M, \rho$) is a left $A$-comodule, and $$ \rho(a m) = a_{(1)}m_{(-1)} \otimes a_{(2)}m_{(0)}.$$

The book uses the convention of Sumless Sweedler notation that

1. $\Delta (a) = a_{(1)} \otimes a_{(2)}$
2. $\rho (m) = m_{(-1)} \otimes m_{(0)}$

I'm just looking to gain some intuition about what the axiom for the $\rho$ map in the definition is trying to convey. It's not clear to me what the meaning of this axiom is.