  The problem of injective coloring in graphs can be revisited through two
different approaches: coloring the two-step graphs and vertex partitioning of
graphs into open packing sets, each of which is equivalent to the injective
coloring problem itself. Taking these facts into account, we observe that the
injective coloring lies between graph coloring and domination theory.
  We make use of these three points of view in this paper so as to investigate
the injective coloring of some well-known graph products. We bound the
injective chromatic number of direct and lexicographic product graphs from
below and above. In particular, we completely determine this parameter for the
direct product of two cycles. We also give a closed formula for the corona
product of two graphs.
