A directed hypergraph (dihypergraph) consists of a set of vertices and a set of hyperarcs, where each hyperarc is partitioned into a head and a tail. Directed hypergraphs are useful in many applications, including the study of chemical reactions or relational databases. We provide asymptotic formulae for the number of directed hypergraphs with given in-degree sequence, out-degree sequence, and with the head and tail sizes of all hyperarcs specified. Our formulae hold when none of the following parameters are too large: the maximum out-degree, the maximum in-degree, the maximum head size and the maximum tail size.