Mutual genericity for product forcing

If $H=G_0\times G_1$ is generic for product forcing over $M$, then $G_1$ is generic over $M[G_0]$. For a name $\dot{D}$ for a dense subset of the second factor, take $p_0\in G_0$ forcing density. Ground-model pairs with first coordinate incompatible with $p_0$, or with first coordinate below $p_0$ forcing the second into $\dot{D}$, form a dense product set. The existential clause of syntactic forcing and the membership clause for a canonical forcing name give the density witnesses. The product filter meets this set, cannot use the incompatible case, and hence its second projection meets ${\dot{D}}^{G_0}$.