  We prove the following conjecture of S. Thomass\'e: for every (potentially
infinite) digraph $ D $ it is possible to iteratively reverse directed cycles
in such a way that the dichromatic number of the final reorientation $ D^{*} $
of $ D $ is at most two and each edge is flipped only finitely many times. In
addition, we guarantee that in every strong component of $ D^{*} $ all the
local edge-connectivities are finite and any edge is reversed at most twice.
