Construction 2.2.2.1.2 (01EF): Construction of the Coproduct of a Family of Sets

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The disjoint union of $\webleft\{ A_{i}\webright\} _{i\in I}$ is the pair $\webleft (\coprod _{i\in I}A_{i},\webleft\{ \mathrm{inj}_{i}\webright\} _{i\in I}\webright )$ consisting of:

The Colimit. The set $\coprod _{i\in I}A_{i}$ defined by
\[ \coprod _{i\in I}A_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (i,x\webright )\in I\times \webleft(\bigcup _{i\in I}A_{i}\webright)\ \middle |\ \text{$x\in A_{i}$}\webright\} . \]
The Cocone. The collection
\[ \webleft\{ \mathrm{inj}_{i} \colon A_{i}\to \coprod _{i\in I}A_{i}\webright\} _{i\in I} \]

of maps given by

\[ \mathrm{inj}_{i}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (i,x\webright ) \]

for each $x\in A_{i}$ and each $i\in I$.

We claim that $\coprod _{i\in I}A_{i}$ is the categorical coproduct of $\webleft\{ A_{i}\webright\} _{i\in I}$ in $\mathsf{Sets}$. Indeed, suppose we have, for each $i\in I$, a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon \coprod _{i\in I}A_{i}\to C$ making the diagram

commute, being uniquely determined by the condition $\phi \circ \mathrm{inj}_{i}=\iota _{i}$ for each $i\in I$ via

\[ \phi \webleft (\webleft (i,x\webright )\webright )=\iota _{i}\webleft (x\webright ) \]

for each $\webleft (i,x\webright )\in \coprod _{i\in I}A_{i}$.