Proof.
It follows immediately that for each $n \geq 0$, the function $\operatorname{id}_{[n]}^{\ast }: S_{n}^{+} \rightarrow S_{n}^{+}$ is the identity map. To prove $(1)$, it will suffice to show that for every pair of composable morphisms $[n''] \xrightarrow {\gamma } [n'] \xrightarrow {\beta } [n]$ in $\operatorname{{\bf \Delta }}$, we have an equality $\gamma ^{\ast } \circ \beta ^{\ast } = ( \beta \circ \gamma )^{\ast }$ of functions from $S_{n}^{+}$ to $S_{n''}^{+}$. Fix an element $(\alpha , \tau ) \in S_{n}^{+}$, where $\alpha : [n] \rightarrow [m]$ is a surjective nondecreasing function and $\tau $ is an element of $S_{m}$. There is a unique commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ [n''] \ar [r]^-{\gamma } \ar [d]^{\alpha ''} & [n'] \ar [d]^{\alpha '} \ar [r]^-{\beta } & [n] \ar [d]^{\alpha } \\ {[m'']} \ar [r]^-{ \gamma ' } & [m'] \ar [r]^-{ \beta ' } & [m] } \]
in the category $\operatorname{{\bf \Delta }}$, where the vertical maps are surjective and the lower horizontal maps are injective. We then compute
\begin{eqnarray*} (\gamma ^{\ast } \circ \beta ^{\ast } )( \alpha , \tau ) & = & \gamma ^{\ast } ( \alpha ', \beta '^{\ast } \tau ) \\ & = & ( \alpha '', \gamma '^{\ast } \beta '^{\ast } \tau ) \\ & = & ( \alpha '', ( \beta ' \circ \gamma ')^{\ast } \tau ) \\ & = & (\beta \circ \gamma )^{\ast } ( \alpha , \tau ), \end{eqnarray*}
which completes the proof of $(1)$.
Assertion $(2)$ is immediate from the definition. Note that if $\beta : [n'] \rightarrow [n]$ is a nondecreasing surjection, then the map $\beta ^{\ast }: S_{n}^{+} \rightarrow S_{n'}^{+}$ is given by the formula $\beta ^{\ast }( \alpha , \tau ) = ( \alpha \circ \beta , \tau )$. It follows that an $n$-simplex $\sigma = (\alpha , \tau )$ of $S_{\bullet }^{+}$ is nondegenerate if and only if $\alpha : [n] \twoheadrightarrow [m]$ is a bijection: that is, if and only if $\sigma $ belongs to the image of $\iota $. Since the image of $\iota $ is closed under face operators (by virtue of $(2)$), we conclude that $S_{\bullet }^{+}$ is braced and that $\iota $ induces an isomorphism of semisimplicial sets $S_{\bullet } \simeq ( S^{+}_{\bullet } )^{\mathrm{nd} }$.
$\square$