AbstractSuppose that a relational system R can be exhausted (in an obvious sense) by a family Rω, ω ϵ Ω, of subrelational systems, each of which can be mapped by a homomorphism onto a subrelational system Sω of a second relational system S. We show that, under suitable finiteness conditions, there is a homomorphism from R into S which finitely agrees with the homomorphisms mapping Rω onto Sω. A similar result holds for isomorphisms