Section 5.12 Coproducts of sets
For any indexed set \(\{U_a\}_{a\in A}\text{,}\) the coproduct or disjoint union is the set
\begin{equation*}
\coprod_{a\in A}U_a\coloneq\bigcup_{a\in A}U_a\times\{a\}\text{.}
\end{equation*}
For every \(b\in A\text{,}\) there is an attached map
\begin{equation*}
\iota_b\colon\fromto{U_b}{\coprod_{a\in A}U_a}
\end{equation*}
given by \(\iota_b(x)=(x,b)\text{,}\) called the inclusion onto the \(b\)-th summand.
The coproduct is really dual to the product. Here's how: for every set \(S\text{,}\) every indexed set \(\{U_a\}_{a\in A}\text{,}\) and every indexed set \(\left\{ f_a\colon U_a \to S \right\}\) of maps, there exists a unique map
\begin{equation*}
f \colon \coprod_{a\in A}U_a \to S
\end{equation*}
such that \(f\circ\iota_a=f_a\text{.}\) Indeed, the map \(f\) is given by the assignment \((x,a) \mapsto f_a(x)\text{.}\)
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