It is the dual notion of product in categories.
Given two objects $A$ and $B$ in a category, their coproduct is another object $C$ together with two morphisms $i_1: A\to C$ and $i_2: B\to C$ that satisfy the following universal property: if $X$ is any other object in the category with morphisms $f_{1}: A \rightarrow X$ and $f_{2}: B \rightarrow X$ then there is an unique morphism $g: P \rightarrow X$ such that $f_1=g \circ i_1$ and $f_2=g \circ i_2$
The coproduct is usually written as $A+B$ and the arrows $i_1, i_2$ are called injections.
The main example of a coproduct is the disjoint union in the category of sets.
It could also be defined for an arbitrary family of objects.
For the relationship between product, coproduct, direct sum and direct product see \textit{Mathematics for Physics, an ilustrated handbook}, page 34.
Not to be confused with semidirect product.
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Author of the notes: Antonio J. Pan-Collantes
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