By the discussion made before, there is a well-defined map st. Indeed, $\text{St}(K)^s$ is a simplicial set generated by $$\begin{aligned}
\{(\sigma,\tau)\in{K\times{K}}\text{ }|\text{ }\exists \text{   }
\sigma^\prime\in{K^s}\text{   }
  \text{such that}\text{   }
\sigma=d_{\nu_1...\nu_q} \sigma ^\prime,
\tau=d_{\mu_1...\mu_p} \sigma ^\prime\\
\text{  }\text{and}\text{   }
(s_{\nu_q...\nu_1}
\sigma,s_{\mu_p...\mu_1} \tau)\in{\delta(K^s\times{K^s})}
\}.\end{aligned}$$

By Definition  \[4.2\], we can put $\operatorname{st}(\sigma,\tau)=(s_{\nu_q...\nu_1}\sigma,s_{\mu_p...\mu_1}\tau)\in{\operatorname{St}(K^s)}$. Clearly $\operatorname{st}$ is an isomorphism since for $(\sigma,\tau)\in{\delta(K^s\times{K^s})}$ and $\sigma^\prime$ as in Definition  \[4.2\], $(\sigma,\tau)\in{K^s\times{K^s}}$ determines an element in $K\times{K}$ by deleting repetitions and this is unique.

[**Remark 2:**]{} The projection on the first factor $\pi_1:S\times{S}\to{S}$ gives a simplicial map $\pi_1:\text{St}_.(S)\to{S}$. Hence, we obtain a prismatic set $P_.\text{St}(S)=P_.(\pi_1)$ as in Example \[3.3\]. Here with $q=q_0+...+q_p$ and $\sigma=s_{\nu_q...\nu_1}\circ{d_{\nu_1...\nu_q}}\sigma^\prime=\mu_{q_0,...,q_p}\bar{\sigma}$, $\tau=s_{\mu_p...\mu_1}\circ{d_{\mu_1...\mu_p}}\sigma^\prime$, we have $$P_p\text{St}(S)_{q_0,...,q_p}=\{(\sigma,\tau,
\bar{\sigma})\in{\text{St}(S)_{q+p}}\times{S_p}\subset{\delta}(S\times{S})_{q+p}\times{S_p}
\text{   }|\text{   } \text{ }\sigma,\tau\text{
 }\text{given above}
\}.$$ That is, $\pi_1(\sigma,\tau)=\mu_{q_0,\dots,q_p}(\bar{\sigma})$, where $\bar{\sigma}=d_{\nu_1...\nu_q}\sigma^\prime\in{S_p}$. So The elements in $P_p\operatorname{St}(S)_{q_0,\dots,q_p}$ are of the form $(\mu_{q_0,\dots,q_p}\bar{\sigma},\tau,\bar{\sigma})$, where $\tau\in{S_q}$. Here explicitly

$$\mu_{q_0,\dots,q_p}={\hat{s}}_{q+p}\circ{s_{({q+p-1})...({q_0+...+q_{p-1}+p}})}...{\hat{s}}_{q_0+q_1+1}
s_{(q_0+q_1)...(q_0+1)}{\hat{s}}_{q_0}s_{(q_0-1)...(0)}.$$

**[Comparison of the two Star Simplicial Sets]{}\[five\]**
==========================================================

We shall now prove that this is closely related to the prismatic set $\bar{P}S$ defined in the previous section.

\[4.4\]

1\) There is a natural **(surjective)** map $$\bar{p}:\bar{P}_.S_.\to{P{\operatorname{St}}_.(S)_.}$$

2\) If $S=K^s$, where $K$ is a simplicial complex, then $\bar{p}$ is an isomorphism.

1\) Take an element $\gamma\in{\bar{P}_pS_{q_0,...,q_p}}=S_{q_0+...+q_p+2p+1}$. Then $\gamma$ and $q_0,...,q_p$ 