This follows from Chapter 9: Preorders, Item 2 and Item 5 of Proposition 9.1.4.1.2.
This is a repetition of Item 2 of Proposition 2.1.3.1.3 and is proved there.
Item 3: Maps From the Punctual Set
The bijection
\[ \Phi _{A}\colon \mathsf{Sets}\webleft (\text{pt},A\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A \]
is given by
\[ \Phi _{A}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\star \webright ) \]
for each $f\in \mathsf{Sets}\webleft (\text{pt},A\webright )$, admitting an inverse
\[ \Phi ^{-1}_{A}\colon A\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}\webleft (\text{pt},A\webright ) \]
given by
\[ \Phi ^{-1}_{A}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}] \]
for each $a\in A$. Indeed, we have
\begin{align*} \webleft [\Phi ^{-1}_{A}\circ \Phi _{A}\webright ]\webleft (f\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}\webleft (\Phi _{A}\webleft (f\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}\webleft (f\webleft (\star \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto f\webleft (\star \webright )]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\text{id}_{\mathsf{Sets}\webleft (\text{pt},A\webright )}\webright ]\webleft (f\webright ) \end{align*}
for each $f\in \mathsf{Sets}\webleft (\text{pt},A\webright )$ and
\begin{align*} \webleft [\Phi _{A}\circ \Phi ^{-1}_{A}\webright ]\webleft (a\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}\webleft (\Phi ^{-1}_{A}\webleft (a\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}\webleft ([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{ev}_{\star }\webleft ([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\text{id}_{A}\webright ]\webleft (a\webright ) \end{align*}
for each $a\in A$, and thus we have
\begin{align*} \Phi ^{-1}_{A}\circ \Phi _{A} & = \text{id}_{\mathsf{Sets}\webleft (\text{pt},A\webright )}\\ \Phi _{A}\circ \Phi ^{-1}_{A} & = \text{id}_{A}. \end{align*}
To prove naturality, we need to show that the diagram
commutes. Indeed, we have
\begin{align*} \webleft [f\circ \Phi _{A}\webright ]\webleft (\phi \webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\Phi _{A}\webleft (\phi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\phi \webleft (\star \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [f\circ \phi \webright ]\webleft (\star \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}\webleft (f\circ \phi \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}\webleft (f_{*}\webleft (\phi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\Phi _{B}\circ f_{*}\webright ]\webleft (\phi \webright ) \end{align*}
for each $\phi \in \mathsf{Sets}\webleft (\text{pt},A\webright )$. This finishes the proof.
Item 4: Maps to the Punctual Set
This follows from the universal property of $\text{pt}$ as the terminal set, Definition 2.1.1.1.1.
\begin{align*} \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ),\\ \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (B,\mathsf{Sets}\webleft (A,C\webright )\webright ), \end{align*}
