We revisit the Rankin–Selberg integral of Shimura type for generic representations of ${\text{SL}}_2\times {\text{GL}}_2$ 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 the local functional equation for the general linear groups but use the Casselman–Shalika formulas for unramified Whittaker functions for ${\text{SL}}_2$ and ${\text{GL}}_2$.