  For any two finite posets $P$ and $Q$, let $\Hom(P,Q)$ be the hom-poset
consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the
collection of all order ideals of $Q$. In this paper, we study some basic
properties of the hom-poset $\Hom(P,Q)$ and prove that $\Hom\big(P,J(Q)\big)$
is a distributive lattice and characterized by \[ \Hom\big(P,J(Q)\big)\cong
J(P^*\times Q), \] where $P^*$ is the dual of $P$. Consequently, we obtain that
$\Hom\big(P,J(Q)\big)$ and $\Hom\big(Q,J(P)\big)$ are dual isomorphic, i.e., \[
\Hom\big(P,J(Q)\big)\cong \Hom^{*}\big(Q,J(P)\big). \] As applications, we
calculate the number of order preserving maps from any poset to the boolean
algebra, and the characteristic polynomial of $\Hom\big(P,J(Q)\big)$.
