The coproduct of $\left\{ A_{i}\right\} _{i\in I}$1 is the coproduct of $\left\{ A_{i}\right\} _{i\in I}$ in $\mathsf{Sets}$ as in 
, .
- 1Further Terminology: Also called the disjoint union of the family $\left\{ A_{i}\right\} _{i\in I}$.
4.2.2 Coproducts of Families of Sets
Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.
Concretely, the disjoint union of $\left\{ A_{i}\right\} _{i\in I}$ is the pair $\webleft (\coprod _{i\in I}A_{i},\left\{ \mathrm{inj}_{i}\right\} _{i\in I}\webright )$ consisting of:
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The Colimit. The set $\coprod _{i\in I}A_{i}$ defined by
\[ \coprod _{i\in I}A_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ \webleft (i,x\webright )\in I\times \left(\bigcup _{i\in I}A_{i}\right)\ \middle |\ \text{$x\in A_{i}$}\right\} . \] -
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The Cocone. The collection
\[ \left\{ \mathrm{inj}_{i} \colon A_{i}\to \coprod _{i\in I}A_{i}\right\} _{i\in I} \]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 $i\in I$.
Proof of Construction 4.2.2.1.2.
We claim that $\coprod _{i\in I}A_{i}$ is the categorical coproduct of $\left\{ A_{i}\right\} _{i\in I}$ in $\mathsf{Sets}$. Indeed, suppose we have, for each $i\in I$, a diagram of the form
\[ \phi \webleft (\webleft (i,x\webright )\webright )=\iota _{i}\webleft (x\webright ) \]
for each $\webleft (i,x\webright )\in \coprod _{i\in I}A_{i}$.
Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.
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Functoriality. The assignment $\left\{ A_{i}\right\} _{i\in I}\mapsto \coprod _{i\in I}A_{i}$ defines a functor
\[ \coprod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\to \mathsf{Sets} \]where
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Action on Objects. For each $\webleft (A_{i}\webright )_{i\in I}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, we have
\[ \left[\coprod _{i\in I}\right]\webleft (\webleft (A_{i}\webright )_{i\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{i\in I}A_{i} \] -
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Action on Morphisms. For each $\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, the action on $\operatorname {\mathrm{Hom}}$-sets
\[ \left(\coprod _{i\in I}\right)_{\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}} \colon \operatorname {\mathrm{Nat}}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )\to \mathsf{Sets}\left(\coprod _{i\in I}A_{i},\coprod _{i\in I}B_{i}\right) \]of $\coprod _{i\in I}$ at $\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ is defined by sending a map
\[ \left\{ f_{i}\colon A_{i}\to B_{i} \right\} _{i\in I} \]in $\operatorname {\mathrm{Nat}}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ to the map of sets
\[ \coprod _{i\in I}f_{i}\colon \coprod _{i\in I}A_{i}\to \coprod _{i\in I}B_{i} \]defined by
\[ \left[\coprod _{i\in I}f_{i}\right]\webleft (i,a\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{i}\webleft (a\webright ) \]for each $\webleft (i,a\webright )\in \coprod _{i\in I}A_{i}$.
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Proof of Proposition 4.2.2.1.3.
Item 1: Functoriality
This follows from , of .
