In this diagram, the top horizontal map we shall call the prismatic triangulation homeomorphism $$\lambda:|P_.(K/L)|\stackrel{\cong}\to{|K|}$$ induced by $$\begin{aligned}
\label{3.1}
\lambda(t,s^0,...,s^p,(\tau,\sigma))=(t_0s^0,...,t_ps^p,\tau)\in{\Delta^{p+q}\times{K_{p+q}}},\end{aligned}$$ where $(t,s,\tau,\sigma)\in{\Delta^p}\times{\Delta^{q_0...q_p}}\times{P_p(K/L)_{q_0...q_p}}$ and $q=q_0+...+q_p$.

[**Note:**]{} If $\stackrel{\circ}{\sigma}$ is an open $p$-simplex in $L$ then $\lambda$ provides a natural trivialization of $|K|_\sigma=\pi^{-1}(\stackrel{\circ}{\sigma})$, that is, a homeomorphism $$\lambda:\stackrel{\circ}{\sigma}\times{|P_p(K/\sigma)}|\stackrel{\approx}\to{|K|_\sigma}.$$

We can generalize this construction to any simplicial map:

\[3.3\] [*Prismatic triangulation of a simplicial map.*]{} Let $f:S_.\to{\bar{S_.}}$ be a simplicial map of simplicial sets and define $P_.(f)$ by $$P_p(f)_{q_0,...,q_p}=\{(\sigma,\bar{\sigma})\in{S_{q_0+...+q_p+p}}\times{{\bar{S}}_p}\text{
  }|\text{   }
f(\sigma)=\mu_{q_0,...,q_p}(\bar{\sigma})\}$$ where the corresponding map $$\mu^{q_0,...,q_p}:\Delta^{q_0+...+q_p+p}\to{\Delta^p}$$ is given by $$\{0,...,q_0|...|q_0+...+q_{p-1}+p,...,q_0+...+q_p+p\}\to{\{0,...,p\}}.$$ By this, we mean that the basis vectors $e_0,...,e_{q_0}$ are mapped to $e_0$, and $e_{q_0+1},...,e_{q_0+q_1+1}$ are mapped to $e_1$ and etc. Explicitly $$\mu_{q_0,...,q_p}=\hat{s}_{q+p}\circ{s_{(q_0+...+q_p+p-1)...(q_0+...+q_{p-1}+p)}}\circ{...}\circ{\hat{s}_{q_0}}\circ{s_{(q_0-1)...(0)}},$$ where the $\hat{s}_i$ are left out and $$s_{(q_0+...+q_i+i-1)...(q_0+...+q_{i-1}+i)}=s_{q_0+...+q_i+i-1}\circ{...}\circ{s_{q_0+...+q_{i-1}+i}},$$ $i=0,...,p$. The boundary maps in the fibre direction $$d_j^i:P_p(f)_{q_0,...,q_p}\to{P_p(f)_{q_0,...,q_i-1,...,q_p}}$$ are inherited from the face operators defined on $S_{q+p}$. Thus $$d_j^i(\sigma,\bar{\sigma})=(d_{q_0+...+q_{i-1}+i+j-1}\sigma,\bar{\sigma}).$$ Similarly the degeneracy maps $s^i_j$ on $P_p(f)_{q_0,...,q_p}$ $$s^i_j:P_p(f)_{q_0,...,q_p}\to{P_p(f)_{q_0,...,q_i+1,...,q_p}}$$ are inherited from the ones on $S_{q+p}$. That is, $$s_j^i(\sigma,\bar{\sigma})=(s_{q_0+...+q_{i-1}+i+j-1}\sigma,\bar{\sigma}).$$ The boundary maps $$d^i:P_p(f)_{q_0,...,q_p}\to{P_{p-1}(f)_{q_0,...,{\hat{q}}_i,...,q_p}}$$ are determined by the boundary maps defined on both $S_{q+p}$ and ${\bar{S}}_p$. Thus $$d^i(\sigma,\bar{\sigma})=(d_{q_0+...+q_{i-1}+i-1}\circ{...}\circ{d_{q_0+...+q_i+i-1}}\sigma,d_i\bar{\sigma}),$$ here the composition of the face operators can be shortly written as $$d_{(q_0+...+q_{i-1}+i-1)...({q_0+...+q_i+i-1})}=d_{q_0+...+q_{i-1}+i-1}\circ{...}\circ{d_{q_0+...+q_i+i-1}}.$$

[**Note:**]{} $P_.(f)$ is a prismatic set, but in general not a strong 