Coincidence points of two set-valued mappings of metric spaces are analyzed. Uniform estimates are obtained for the distance to the set of coincidence points and to the set of intersections of the graphs of two set-valued mappings. Sufficient conditions for the existence of double fixed points are derived as a consequence of the results obtained. In addition, estimates are obtained for the distance between the sets of coincidence points of two pairs of set-valued mappings. Copyright © by SIAM. © 2015 Society for Industrial and Applied Mathematics.