  A digraph is $3$-dicritical if it cannot be vertex-partitioned into two sets
inducing acyclic digraphs, but each of its proper subdigraphs can. We give a
human-readable proof that the number of 3-dicritical semi-complete digraphs is
finite. Further, we give a computer-assisted proof of a full characterization
of 3-dicritical semi-complete digraphs. There are eight such digraphs, two of
which are tournaments. We finally give a general upper bound on the maximum
number of arcs in a $3$-dicritical digraph.
