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 _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.