  In this note, we revisit the Rankin-Selberg integral of Shimura type for
generic representations of $\mathrm{SL}_2\times \mathrm{GL}_2$, constructed by
Ginzburg, Rallis, and Soudry. We give a different and more ``intrinsic'' proof
of the unramified computation. In contrast to their proof we avoid local
functional equation for the general linear groups but use the Casselman-Shalika
formulas for unramified Whittaker functions for $\mathrm{SL}_2$ and
$\mathrm{GL}_2$.
