  According to Walter M. Fitch, two genes are xenologs if they are separated by
at least one horizontal gene transfer. This concept is formalized through Fitch
relations, which are defined as binary relations that comprise all pairs
$(x,y)$ of genes $x$ and $y$ for which $y$ has been horizontally transferred at
least once since it diverged from the least common ancestor of $x$ and $y$.
This definition, in particular, preserves the directional character of the
transfer. Fitch relations are characterized by a small set of forbidden induced
subgraphs on three vertices and can be recognized in linear time.
  In this contribution, we provide two novel characterizations of Fitch
relations and present an alternative, short and elegant proof of the
characterization theorem established by Gei{\ss} et al.\ in \emph{J.\ Math.\
Bio 77(5), 2018}.
