  In this paper we provide explicit dual Ramsey statements for several classes
of finite relational structures (such as finite linearly ordered graphs, finite
linearly ordered metric spaces and finite posets with a linear extension) and
conclude the paper with an explicit dual of the Ne\v{s}et\v{r}il-R\"odl Theorem
for relational structures. Instead of embeddings which are crucial for "direct"
Ramsey results, for each class of structures under consideration we propose a
special class of surjective maps and prove a dual Ramsey theorem in such a
setting. In contrast to on-going Ramsey classification projects where the
research is focused on fine-tuning the objects, in this paper we advocate the
idea that fine-tuning the morphisms is the key to proving dual Ramsey results.
Since the setting we are interested in involves both structures and morphisms,
all our results are spelled out using the reinterpretation of the (dual) Ramsey
property in the language of category theory.
