Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.
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Functoriality. The assignments
\[ \webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )\mapsto \webleft (X\vee Y,p_{0}\webright ) \]
define functors
\begin{align*} X\vee - & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -\vee Y & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -_{1}\vee -_{2} & \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{align*} -
Associativity. We have an isomorphism of pointed sets
\[ \webleft (X\vee Y\webright )\vee Z\cong X\vee \webleft (Y\vee Z\webright ), \]
natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \mathsf{Sets}_{*}$.
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Unitality. We have isomorphisms of pointed sets
\begin{align*} \webleft (\text{pt},*\webright )\vee \webleft (X,x_{0}\webright ) & \cong \webleft (X,x_{0}\webright ),\\ \webleft (X,x_{0}\webright )\vee \webleft (\text{pt},*\webright ) & \cong \webleft (X,x_{0}\webright ),\end{align*}
natural in $\webleft (X,x_{0}\webright )\in \mathsf{Sets}_{*}$.
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Commutativity. We have an isomorphism of pointed sets
\[ X\vee Y \cong Y\vee X, \]
natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \mathsf{Sets}_{*}$.
- Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\vee ,\text{pt}\webright )$ is a symmetric monoidal category.
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The Fold Map. We have a natural transformation called the fold map, whose component
\[ \nabla _{X} \colon X\vee X \to X \]at $X$ is given by
\[ \nabla _{X}\webleft (p\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} x & \text{if $p=\webleft [\webleft (0,x\webright )\webright ]$,}\\ x & \text{if $p=\webleft [\webleft (1,x\webright )\webright ]$} \end{cases} \]for each $p\in X\vee X$.
