  Finite digraphs $R$ and $S$ are studied with $\# {\cal H}(G,R) \leq \# {\cal
H}(G,S)$ for every finite digraph $G \in \mathfrak{ D }'$, where ${\cal
H}(G,H)$ is the set of order homomorphisms from $G$ to $H$ and $\mathfrak{ D
}'$ is a class of finite digraphs. It is shown that for several classes
$\mathfrak{ D }'$ of digraphs and $R \in \mathfrak{ D }'$, the relation $\#
{\cal H}(G,R) \leq \# {\cal H}(G,S)$ for every $G \in \mathfrak{ D }'$ is
implied by the relation $\# {\cal S}(G,R) \leq \# {\cal S}(G,S)$ for every $G
\in \mathfrak{ D }'$, where ${\cal S}(G,H)$ is the set of homomorphisms from
$G$ to $H$ mapping all proper arcs of $G$ to proper arcs of $H$. Under an
application-oriented regularity condition, the two relations are even
equivalent. A method is developed for the rearrangement of a digraph $R$,
resulting in a digraph $S$ with $\# {\cal H}(G,R) \leq \# {\cal H}(G,S)$ for
every digraph $G$. The method is applied in constructing pairs of partially
ordered sets $R$ and $S$ with $\# {\cal H}(P,R) \leq \# {\cal H}(P,S)$ for
every partially ordered set $P$. The main part of the results holds also for
undirected graphs.
