  A structure is called homogeneous if every isomorphism between finitely
induced substructures of the structure extends to an automorphism of the
structure. Recently, P. J. Cameron and J. Ne\v{s}et\v{r}il introduced a relaxed
version of homogeneity: we say that a structure is homomorphism-homogeneous if
every homomorphism between finitely induced substructures of the structure
extends to an endomorphism of the structure.
  In this paper we consider finite homomorphism-homogeneous relational systems
with one reflexive binary relation. We show that for a large part of such
relational systems (bidirectionally connected digraphs; a digraph is
bidirectionally connected if each of its connected components can be traversed
by $\rightleftarrows$-paths) the problem of deciding whether the system is
homomorphism-homogeneous is coNP-complete. Consequently, for this class of
relational systems we cannot hope for a description involving a catalogue,
where by a catalogue we understand a finite list of polynomially decidable
classes of structures. On the other hand, in case of bidirectionally
disconnected digraphs we present the full characterization. Our main result
states that if a digraph is bidirectionally disconnected, then it is
homomorphism-homogeneous if and only if it is either a finite
homomorphism-homogeneous quasiorder, or an inflation of a
homomorphism-homogeneous digraph with involution (a peculiar class of digraphs
introduced later in the paper), or an inflation of a digraph whose only
connected components are $C_3^\circ$ and $\1^\circ$.
