Construction 2.2.3.1.2 (01EJ): Construction of Coproducts of Sets

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The coproduct of $A$ and $B$ is the pair $\webleft (A\coprod B,\webleft\{ \mathrm{inj}_{1},\mathrm{inj}_{2}\webright\} \webright )$ consisting of:

The Colimit. The set $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ defined by
\begin{align*} A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{z\in \webleft\{ A,B\webright\} }z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (0,a\webright )\in S\ \middle |\ a\in A\webright\} \cup \webleft\{ \webleft (1,b\webright )\in S\ \middle |\ b\in B\webright\} , \end{align*}

where $S=\webleft\{ 0,1\webright\} \times \webleft (A\cup B\webright )$.
The Cocone. The maps
\begin{align*} \mathrm{inj}_{1} & \colon A \to A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B,\\ \mathrm{inj}_{2} & \colon B \to A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B, \end{align*}

given by

\begin{align*} \mathrm{inj}_{1}\webleft (a\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (0,a\webright ),\\ \mathrm{inj}_{2}\webleft (b\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (1,b\webright ), \end{align*}

for each $a\in A$ and each $b\in B$.

We claim that $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ is the categorical coproduct of $A$ and $B$ in $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\to C$ making the diagram

commute, being uniquely determined by the conditions

\begin{align*} \phi \circ \mathrm{inj}_{A} & = \iota _{A},\\ \phi \circ \mathrm{inj}_{B} & = \iota _{B} \end{align*}

via

\[ \phi \webleft (x\webright )=\begin{cases} \iota _{A}\webleft (a\webright ) & \text{if $x=\webleft (0,a\webright )$,}\\ \iota _{B}\webleft (b\webright ) & \text{if $x=\webleft (1,b\webright )$} \end{cases} \]

for each $x\in A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$.