What is the coproduct of structures?

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The coproduct of a family of L-structures, where L consists of constant, relation, and function symbols, does exist. The coproduct is defined as the disjoint union of the structures, equipped with appropriate relations and functions that respect the original structures. The maps from each element of the family S to the coproduct are indeed monic, meaning they are injective. For further understanding, references to category theory and model theory are essential.

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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?

If you are aware of references related to this question, I would be happy to know them.

Thanks in advance for your comments.
 
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