  Based on an idea of Y. P\'eresse and some results of Maltcev, Mitchell and
Ru\v{s}kuc, we present sufficient conditions under which the endomorphism
monoid of a countably infinite ultrahomogeneous first-order structure has the
Bergman property. This property has played a prominent role both in the theory
of infinite permutation groups and, more recently, in semigroup theory. As a
byproduct of our considerations, we establish a criterion for a countably
infinite ultrahomogeneous structure to be homomorphism-homogeneous.
