Let $\webleft (X,x_{0}\webright )$ be a pointed set.
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Functoriality. The assignment $\webleft (X,x_{0}\webright )\mapsto X^{-}$ defines a functor
\[ X^{-}\colon \mathsf{Sets}^{\mathrm{actv}}_{*}\to \mathsf{Sets}. \]
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Adjointness. We have an adjunction witnessed by a bijection of sets
\begin{align*} \mathsf{Sets}\webleft (X^{-},Y\webright )\cong \mathsf{Sets}_{*}\webleft (X,Y^{+}\webright ),\end{align*}natural in $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ and $Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Symmetric Strong Monoidality With Respect to Wedge Sums. The functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{-},\webleft (-\webright )^{-,\vee },\webleft (-\webright )^{-,\vee }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathrm{actv}}_{*},\vee ,\text{pt}\webright ), \to \webleft (\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\text{Ø}\webright ), \]
being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{-,\vee }_{X,Y} \colon X^{-}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\vee Y\webright )^{-},\\ \webleft (-\webright )^{-,\vee }_{\mathbb {1}} \colon \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{pt}^{-}, \end{gathered} \]natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
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Symmetric Strong Monoidality With Respect to Smash Products. The free pointed set functor of Item 1 has a symmetric strong monoidal structure
\[ \webleft (\webleft (-\webright )^{-},\webleft (-\webright )^{-,\times },\webleft (-\webright )^{-,\times }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets}^{\mathrm{actv}}_{*},\wedge ,S^{0}\webright ), \to \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \]
being equipped with isomorphisms of pointed sets
\[ \begin{gathered} \webleft (-\webright )^{-,\times }_{X,Y} \colon X^{-}\times Y^{-} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (X\wedge Y\webright )^{-},\\ \webleft (-\webright )^{-,\times }_{\mathbb {1}} \colon \text{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (S^{0}\webright )^{-}, \end{gathered} \]natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
