Coproduct

A dual notion to the Product. Let and be Objects of a Category . Their coproduct consists of maps and such that for any and there is a unique making the following a Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
& X
\arrow[dd, "\tilde{f}"]
\\
\\
& A + B
\\
A
\arrow[ur, swap, "\nu_{1}"]
\arrow[uuur, "f_{1}"]
 & & 
 B
 \arrow[ul, "\nu_{2}"]
 \arrow[uuul, swap, "f_{2}"]
 
\end{tikzcd}
\end{document}

If is Locally Small we can have a Functor defined by

If is Representable, with Representation then its Universal Element is some where and and this is exactly the coproduct.

In the Category of Sets, the coproduct is the disjoint union

together with maps and .