• Morphisms From the Monoidal Unit. We have a bijection of sets2
    \[ \mathsf{Sets}_{*}\webleft (S^{0},X\webright ) \cong X, \]

    natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, internalising also to an isomorphism of pointed sets

    \[ \textbf{Sets}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ), \]

    again natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.


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