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In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph G with minimum degree at least 2 is a proper vertex coloring of G such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of G is called the double total dominator chromatic number, denoted by χ d d t (G). Therefore, we establish the close relationship between the double total dominator chromatic number χ d d t (G) and the double total domination number γ × 2 , t (G). We prove the NP-completeness of the problem. We also examine the effects on χ d d t (G) when G is modified by some operations. Finally, we discuss the χ d d t (G) number of square of trees by giving some bounds. [ABSTRACT FROM AUTHOR]