The disjoint union of the family $\webleft\{ A_{i}\webright\} _{i\in I}$ 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 $i\in I$.