  To a branched cover f between orientable surfaces one can associate a certain
branch datum D(f), that encodes the combinatorics of the cover. This D(f)
satisfies a compatibility condition called the Riemann-Hurwitz relation. The
old but still partly unsolved Hurwitz problem asks whether for a given abstract
compatible branch datum D there exists a branched cover f such that D(f)=D. One
can actually refine this problem and ask how many these f's exist, but one must
of course decide what restrictions one puts on such f's, and choose an
equivalence relation up to which one regards them. And it turns out that quite
a few natural choices are possible. In this short note we carefully analyze all
these choices and show that the number of actually distinct ones is only three.
To see that these three choices are indeed different we employ Grothendieck's
dessins d'enfant.
