This follows from Chapter 11: Categories, Item 2 and Item 5 of Proposition 11.1.4.1.2.
This is a repetition of Item 2 of Proposition 4.1.3.1.3 and is proved there.
Item 3: Maps From the Punctual Set
The bijection
\[ \Phi _{A}\colon \mathsf{Sets}(\mathrm{pt},A)\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A \]
is given by
\[ \Phi _{A}(f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(\star ) \]
for each $f\in \mathsf{Sets}(\mathrm{pt},A)$, admitting an inverse
\[ \Phi ^{-1}_{A}\colon A\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}(\mathrm{pt},A) \]
given by
\[ \Phi ^{-1}_{A}(a)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}] \]
for each $a\in A$. Indeed, we have
\begin{align*} [\Phi ^{-1}_{A}\circ \Phi _{A}](f) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}(\Phi _{A}(f))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}(f(\star ))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto f(\star )]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\operatorname {\mathrm{id}}_{\mathsf{Sets}(\mathrm{pt},A)}](f) \end{align*}
for each $f\in \mathsf{Sets}(\mathrm{pt},A)$ and
\begin{align*} [\Phi _{A}\circ \Phi ^{-1}_{A}](a) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}(\Phi ^{-1}_{A}(a))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}])\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{ev}_{\star }([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}])\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\operatorname {\mathrm{id}}_{A}](a) \end{align*}
for each $a\in A$, and thus we have
\begin{align*} \Phi ^{-1}_{A}\circ \Phi _{A} & = \operatorname {\mathrm{id}}_{\mathsf{Sets}(\mathrm{pt},A)}\\ \Phi _{A}\circ \Phi ^{-1}_{A} & = \operatorname {\mathrm{id}}_{A}. \end{align*}
To prove naturality, we need to show that the diagram
commutes. Indeed, we have
\begin{align*} [f\circ \Phi _{A}](\phi ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(\Phi _{A}(\phi ))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f(\phi (\star ))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[f\circ \phi ](\star )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}(f\circ \phi )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}(f_{!}(\phi ))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\Phi _{B}\circ f_{!}](\phi ) \end{align*}
for each $\phi \in \mathsf{Sets}(\mathrm{pt},A)$. This finishes the proof.
Item 4: Maps to the Punctual Set
This follows from the universal property of $\mathrm{pt}$ as the terminal set, Definition 4.1.1.1.1.
