Let D=(V, A) be a finite simple directed graph (shortly, digraph). A function f: V→{− 1, 0, 1} is called a twin minus total dominating function (TMTDF) if f (N−(v))≥ 1 and f (N+(v))≥ 1 for each vertex v∈ V. The twin minus total domination number of D is y*mt(D)= min {w (f)| f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for y*mt(D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs
