Item 4 (00A0) — The Clowder Project

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*To be replaced with Linus Romer's Elemaints when it is released.

Morphisms From the Monoidal Unit. We have a bijection of sets ^[1]
\[ \mathsf{Sets}_{*}\webleft (S^{0},X\webright ) \cong X, \]

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

Footnotes

[1] In other words, the forgetful functor

\[ {\text{忘}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets} \]

defined on objects by sending a pointed set to its underlying set is corepresentable by $S^{0}$.

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