  In this paper, we state and prove two Fra\"{i}ss\'{e}-style results that
cover existence and uniqueness properties for twelve of the eighteen different
notions of homomorphism-homogeneity as introduced by Lockett and Truss, and
provide forward directions and implications for the remaining six cases.
Following these results, we completely determine the extent to which the
countable homogeneous undirected graphs (as classified by Lachlan and Woodrow)
are homomorphism-homogeneous; we also provide some insight into the directed
graph case.
