  We prove that the intersection hypergraph of a family of $n$ pseudo-disks
with respect to another family of pseudo-disks admits a proper coloring with
$4$ colors and a conflict-free coloring with $O(\log n)$ colors. Along the way
we prove that the respective Delaunay-graph is planar. We also prove that the
intersection hypergraph of a family of $n$ regions with linear union complexity
with respect to a family of pseudo-disks admits a proper coloring with
constantly many colors and a conflict-free coloring with $O(\log n)$ colors.
Our results serve as a common generalization and strengthening of many earlier
results, including ones about proper and conflict-free coloring points with
respect to pseudo-disks, coloring regions of linear union complexity with
respect to points and coloring disks with respect to disks.
