Let $[n]=\{0,\dots,n-1\}$, $[0]=\emptyset$ and $\Sigma_{mn}$ be a set of all maps from $[m]$ to $[n]$, ($\Sigma_{0n}$ consists of a single map and $\Sigma_{no}=\emptyset$ for all $n>0$). Also let $[N]=\{[n]\}$, $\Sigma =\bigcup\Sigma_{mn}$ and $S=\langle[N],\Sigma\rangle$.
Now consider a "pseudo-diagramme" in Sets corresponding to $S$, i.e. a family of (non-empty) sets $(D_n)_{n>0}$ and a family of maps $\pi=(\pi_\sigma)_{\sigma\in \Sigma}$, $\pi_\sigma:D_n\to D_m$ for $\sigma\in \Sigma_{mn}$ (composition and identity may be not preserved).
This kind of structures are closely related to simplicial sets where instead of arbitrary maps are taken only monotone maps and composition and identity map are preserved. So a structure from above (without level $0$) can be regarded as a simplicial set with extra maps $\pi_\sigma$ corresponding to permutations $[n]\to[n]$.
I am looking for reference on such structures in literature, which should exist already somewhere in mathematical practice.