Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual Ramsey theory. The main goal of this paper is to advance the dual Ramsey theory for finite relational structures with respect to natural structure-preserving maps. Tools from category theory prove instrumental in this endeavor, as was previously the case for finite algebraic systems where the dual Ramsey property had been established for every class of finite algebras coming from an equationally defined class. One cannot help but feel that dual Ramsey phenomena are deeply connected to categorical strategies.