The equivalent way of writing the Yetter-Drinfeld compatibility condition between left action and left coaction described in this post can be easily translated into a commutative diagram. It looks like this $$ (\nabla\otimes\rho)(1\otimes\tau\otimes 1)(1\otimes\delta)=(\nabla^\mathrm{op}\otimes 1)(1\otimes\delta)(1\otimes\rho)(\Delta^\mathrm{op}\otimes 1)\,. $$ Similary equations work for ${}_H\mathcal{YD}^H$ instead of ${}_H^H\mathcal{YD}$, see for example Kassel, Definition IX.5.1, page 220. These equations tell us how $\rho$ and $\delta$ "commute". We need precisely this condition to compute the Drinfeld center of ${}_H\mathrm{Mod}$.

The equivalent way of writing the Yetter-Drinfeld compatibility condition between left action and left coaction described in this post can be easily translated into a commutative diagram. It looks like this $$ (\nabla\otimes\rho)(1\otimes\tau\otimes 1)(\Delta\otimes\delta)=(\nabla^\mathrm{op}\otimes 1)(1\otimes\delta)(1\otimes\rho)(\Delta^\mathrm{op}\otimes 1)\,. $$ Similary equations work for ${}_H\mathcal{YD}^H$ instead of ${}_H^H\mathcal{YD}$, see for example Kassel, Definition IX.5.1, page 220. These equations tell us how $\rho$ and $\delta$ "commute". We need precisely this condition to compute the Drinfeld center of ${}_H\mathrm{Mod}$.