The associator of the coproduct of sets is the natural isomorphism
\[ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}} \colon \mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft (\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\circ {\webleft (\text{id}_{\mathsf{Sets}}\times \mathord {\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}}, \]
as in the diagram
whose component
\[ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z} \colon \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright ) \]
at $\webleft (X,Y,Z\webright )$ is given by
\begin{align*} \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft (0,x\webright ) & \text{if $a=\webleft (0,\webleft (0,x\webright )\webright )$,}\\ \webleft (1,\webleft (0,y\webright )\webright ) & \text{if $a=\webleft (0,\webleft (1,y\webright )\webright )$,}\\ \webleft (1,\webleft (1,a\webright )\webright ) & \text{if $a=\webleft (1,z\webright )$} \end{cases}\end{align*}
for each $a\in \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z$.
Unwinding the Definitions of $\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z$ and $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )$
Firstly, we unwind the expressions for $\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z$ and $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )$. We have
\begin{align*} \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (0,a\webright )\in S\ \middle |\ a\in X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright\} \cup \webleft\{ \webleft (1,z\webright )\in S\ \middle |\ z\in Z\webright\} \\ & = \webleft\{ \webleft (0,\webleft (0,x\webright )\webright )\in S\ \middle |\ x\in X\webright\} \cup \webleft\{ \webleft (0,\webleft (1,y\webright )\webright )\in S\ \middle |\ y\in Y\webright\} \\ & \phantom{={}} \mkern 4mu\cup \webleft\{ \webleft (1,z\webright )\in S\ \middle |\ z\in Z\webright\} , \end{align*}
where $S=\webleft\{ 0,1\webright\} \times \webleft (\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\cup Z\webright )$ and
\begin{align*} X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (0,x\webright )\in S'\ \middle |\ x\in X\webright\} \cup \webleft\{ \webleft (1,a\webright )\in S'\ \middle |\ a\in Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright\} \\ & = \webleft\{ \webleft (0,x\webright )\in S'\ \middle |\ x\in X\webright\} \cup \webleft\{ \webleft (1,\webleft (0,y\webright )\webright )\in S'\ \middle |\ y\in Y\webright\} \\ & \phantom{={}} \mkern 4mu\cup \webleft\{ \webleft (1,\webleft (1,z\webright )\webright )\in S’\ \middle |\ z\in Z\webright\} , \end{align*}
where $S'=\webleft\{ 0,1\webright\} \times \webleft (X\cup \webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )\webright )$.
Invertibility
The inverse of $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z}$ is the map
\[ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y,Z}\colon X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )\to \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z \]
given by
\begin{align*} \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y,Z}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft (0,\webleft (0,x\webright )\webright ) & \text{if $a=\webleft (0,x\webright )$,}\\ \webleft (0,\webleft (1,y\webright )\webright ) & \text{if $a=\webleft (1,\webleft (0,y\webright )\webright )$,}\\ \webleft (1,z\webright ) & \text{if $a=\webleft (1,\webleft (1,z\webright )\webright )$} \end{cases}\end{align*}
for each $a\in X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webleft (\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )$. Indeed:
- Invertibility I. The map $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z}$ acts on elements as
\begin{align*} & \mkern 40mu\mathclap {\webleft (0,\webleft (0,x\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,x\webright )} \mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,\webleft (0,x\webright )\webright )}\mkern 35mu\mathrlap {,}\\ & \mkern 40mu\mathclap {\webleft (0,\webleft (0,y\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,\webleft (0,y\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,\webleft (0,y\webright )\webright )}\mkern 35mu\mathrlap {,}\\ & \mkern 40mu\mathclap {\webleft (1,z\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,\webleft (1,z\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,z\webright )} \end{align*}
and hence is equal to the identity map of $\webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z$.
- Invertibility II. The map $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z}\circ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}_{X,Y,Z}$ acts on elements as
\begin{align*} & \mkern 40mu\mathclap {\webleft (0,x\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,\webleft (0,x\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,x\webright )}\mkern 20mu\mathrlap {,}\\ & \mkern 40mu\mathclap {\webleft (1,\webleft (0,y\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (0,\webleft (0,y\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,\webleft (0,y\webright )\webright )}\mkern 35mu\mathrlap {,}\\ & \mkern 40mu\mathclap {\webleft (1,\webleft (1,z\webright )\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,z\webright )}\mkern 40mu \mapsto \mkern 40mu\mathclap {\webleft (1,\webleft (1,z\webright )\webright )}\mkern 40mu \end{align*}
and hence is equal to the identity map of $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright )$.
Therefore $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z}$ is indeed an isomorphism.
Naturality
We need to show that, given functions
\begin{align*} f & \colon X \to X',\\ g & \colon Y \to Y',\\ h & \colon Z \to Z’ \end{align*}
the diagram
commutes. Indeed, this diagram acts on elements as
and hence indeed commutes, showing $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$ to be a natural transformation.
Being a Natural Isomorphism
Since $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$ is natural and $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},-1}$ is a componentwise inverse to $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, it follows from Chapter 9: Categories, Item 2 of Proposition 9.9.7.1.2 that $\lambda ^{\mathsf{Sets},-1}$ is also natural. Thus $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$ is a natural isomorphism.
