  The dichromatic number $\vec\chi(D)$ of a digraph $D$ is the minimum size of
a partition of its vertices into acyclic induced subgraphs. We denote by
$\lambda(D)$ the maximum local edge connectivity of a digraph $D$. Neumann-Lara
proved that for every digraph $D$, $\vec\chi(D) \leq \lambda(D) + 1$. In this
paper, we characterize the digraphs $D$ for which $\vec\chi(D) = \lambda(D) +
1$. This generalizes an analogue result for undirected graph proved by Stiebitz
and Toft as well as the directed version of Brooks' Theorem proved by Mohar.
Along the way, we introduce a generalization of Haj\'os join that gives a new
way to construct families of dicritical digraphs that is of independent
interest.
