Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

  1. Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \textbf{Sets}_{*}\webleft (X,Y\webright )$ define functors
    \[ \begin{array}{ccc} \textbf{Sets}_{*}\webleft (X,-\webright )\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \textbf{Sets}_{*}\webleft (-,Y\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \textbf{Sets}_{*}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]

    In particular, given pointed maps

    \begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (A,a_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (B,b_{0}\webright ), \end{align*}

    the induced map

    \[ \textbf{Sets}_{*}\webleft (f,g\webright )\colon \textbf{Sets}_{*}\webleft (A,Y\webright )\to \textbf{Sets}_{*}\webleft (X,B\webright ) \]

    is given by

    \[ \webleft [\textbf{Sets}_{*}\webleft (f,g\webright )\webright ]\webleft (\phi \webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \circ f \]

    for each $\phi \in \textbf{Sets}_{*}\webleft (A,Y\webright )$.

  2. Adjointness. We have adjunctions
    witnessed by bijections
    \begin{align*} \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\textbf{Sets}_{*}\webleft (Y,Z\webright )\webright ),\\ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\textbf{Sets}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

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

  3. Enriched Adjointness. We have $\mathsf{Sets}_{*}$-enriched adjunctions
    witnessed by isomorphisms of pointed sets
    \begin{align*} \textbf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \textbf{Sets}_{*}\webleft (X,\textbf{Sets}_{*}\webleft (Y,Z\webright )\webright ),\\ \textbf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \textbf{Sets}_{*}\webleft (X,\textbf{Sets}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

    natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\textbf{Sets}_{*}\webright )$.

Item 1: Functoriality
This follows from Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.5.1.2 and from the equalities
\begin{align*} g\circ \Delta _{y_{0}} & = \Delta _{z_{0}},\\ \Delta _{y_{0}}\circ f & = \Delta _{y_{0}} \end{align*}

for morphisms $f\colon \webleft (K,k_{0}\webright )\to \webleft (X,x_{0}\webright )$ and $g\colon \webleft (Y,y_{0}\webright )\to \webleft (Z,z_{0}\webright )$, which guarantee pre- and postcomposition by morphisms of pointed sets to also be morphisms of pointed sets.

Item 2: Adjointness
This is a repetition of Item 2 of Proposition 5.5.1.1.10, and is proved there.
Item 3: Enriched Adjointness
This is a repetition of Item 3 of Proposition 5.5.1.1.10, and is proved there.


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