  In this paper we show that a countable structure admitting a finite
monomorphic decomposition has finite big Ramsey degrees if and only if so does
every monomorphic part in its minimal monomorphic decomposition. The necessary
prerequisite for this result is the characterization of monomorphic structures
with finite big Ramsey degrees: a countable monomorphic structure has finite
big Ramsey degrees if and only if it is chainable by a chain with finite big
Ramsey degrees. Interestingly, both characterizations require deep structural
properties of chains. Fra\"{i}ss\'{e}'s Conjecture (actually, its positive
resolution due to Laver) is instrumental in the characterization of monomorphic
structures with finite big Ramsey degrees, while the analysis of big Ramsey
combinatorics of structures admitting a finite monomorphic decomposition
requires a product Ramsey theorem for big Ramsey degrees. We find this last
result particularly intriguing because big Ramsey degrees misbehave notoriously
when it comes to general product statements.
