For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \otimes m^{(0)}$. For $M$ to be a Yetter-Drinfeld module, it must satisfy the compatibility condition $$ \delta(\rho(h,m)) = h_{(1)}m^{(-1)} S(h_{(3)}) \otimes \rho(h_{(2)},m^{(0)})\,. $$ Is there a nice commutative diagram that can be drawn to illustrate this compatibility condition? I've included my best attempt below in a CW answer. Is there more reason behind this condition besides "it's the condition we need to be true for the nice braiding to work out in the Yetter-Drinfeld category?".

Note also this post and this post. Looking at alternative characterizations of these modules might help.

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \otimes m^{(0)}$. For $M$ to be a Yetter-Drinfeld module, it must satisfy the compatibility condition $$ \delta(\rho(h,m)) = h_{(1)}m^{(-1)} S(h_{(3)}) \otimes \rho(h_{(2)},m^{(0)})\,. $$ Is there a natural commutative diagram to draw that illustrates this compatibility condition? I've included my attempt below in a CW answer. Also, is there any more reason behind this condition besides "it's the condition we need to be true to get the nice braiding to work out in the Yetter-Drinfeld category?".