Suppose $S$ is a family of $L$-structures where $L$ is some collection of constant symbols, relation symbols, and function symbols. Does the coproduct of elements of $S$ exist?
If not, how does one prove it?
If yes, how is the coproduct defined? Are the maps from elements of S to the coproduct all monic?
Also, any references speaking about this would be appreciated; something that involves mathematical logic and category theory perhaps.
Thanks in advance for your answer.