We consider the properties and objects assosiated with a finite relational structure H and listed in the title: - structure H has finite duality - for any G the set of homomorphisms hom(G,H) is connected, that is, any homomorphism can be transformed to any other homomrphism through a sequence of homomorphisms such that any two consequent ones only differ in bounded number of points - for any G the set hom(G,H) satisfies the strong spatial mixing property defined here in terms of extedability of partial homomorphisms - for any G there exist parameters (activities) such that hom(G,H) has a unique Gibbs measure. We show that for any H these properties are closely related.