  Big Ramsey degrees of finite structures are usually considered with respect
to a Fra\"{i} ss\'e limit. Building mainly on the work of Devlin, Sauer,
Laflamme and Van Th\'e, in this paper we consider structures which are not
Fra\"{i} ss\'e limits, and still have the property that their finite
substructures have finite big Ramsey degrees in them. For example, the class of
all finite acyclic oriented graphs is not a Fra\"{i} ss\'e class, and yet we
show that there is a countably infinite acyclic oriented graph in which every
finite acyclic oriented graph has finite big Ramsey degree. Our main tools come
from category theory as it has recently become evident that the Ramsey property
is not only a deep combinatorial property, but also a genuine categorical
property.
