A container theorem for general digraphs with forbidden subdigraphs

In a seminal work, K\"uhn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle $DK_3$. In this paper we prove a container theorem for general digraphs under a natural sparsity condition. For the edge-weight parameter $a=2$, this condition permits digraphs with $2$-cycles (density at most $1$) but excludes denser obstructions like $DK_3$; for larger $a$ it allows digraphs with a controlled density of $2$-cycles. As applications, we obtain asymptotic counting results for $H$-free digraphs and describe the typical structure of digraphs avoiding a fixed digraph $H$ satisfying our condition. Our results unify and extend several previous results in the area.