  A structure is called weakly oligomorphic if it realizes only finitely many
n-ary positive existential types for every n. The goal of this paper is to show
that the notions of homomorphism-homogeneity, and weak oligomorphy are not only
completely analogous to the classical notions of ultrahomogeneity and
oligomorphy, but are actually closely related. A first result is a
Fra\"iss\'e-type theorem for homomorphism-homogeneous relational structures.
Further we show that every weakly oligomorphic homomorphism-homogeneous
structure contains (up to isomorphism) a unique homogeneous,
homomorphism-homogeneous core, to which it is homomorphism-equivalent. As a
consequence, we obtain that every countable weakly oligomorphic structure is
homomorphism-equivalent with a finite or \omega-categorical structure. Another
result is the characterization of positive existential theories of weakly
oligomorphic structures as the positive existential parts of \omega-categorical
theories. Finally, we show, that the countable models of countable weakly
oligomorphic structures are mutually homomorphism-equivalent (we call first
order theories with this property weakly \omega-categorical). These results are
in analogy with part of the Engeler-Ryll-Nardzewski-Svenonius-theorem.
