One uses the following trick.

By the projection formula we have
$${p_2}_*(\mathcal{O}_{X\times Y_1}) \cong {p_2}_*(p_2^*M_1^{-1} \otimes L_1) \cong M_1^{-1} \otimes {p_2}_*(L_1)$$
and since $X$ is complete (and $k$ is algebraically closed) it follows from the Künneth formula that 
$${p_2}_*\mathcal{O}_{X\times Y_1} \cong \mathcal{O}_{Y_1}$$
which gives the desired isomorphism on $Y_1$ by uniqueness of inverses up to isomorphism in the Picard group.