For any $n\in\mathbb{N}$ and for any $i\in[n]:=\{1,\dots,n\}$, you may consider the maps $\tau_i:[n-1] \to [n]$ defined in Iverson notation by $$\tau_i(x):=x+[x\ge i]\\ .$$ That is, $\tau_i(x)=x$ unless $x\ge i$, in which case it is $x+1$). It induces by composition the map $\tau_i^*:\mathbb{N}^n\to\mathbb{N}^{n-1}$ that takes the element $x\in \mathbb{N}^n$ to $x\circ\tau_i$, which is what you want. In case of need, to recall the domain we may write $\tau_{i,n}$ instead of $\tau_i$; also, to simplify the notation, $\tau_i\cdot x$ instead of $\tau_i^*(x)$.

This and similar notations are somehow useful e.g. in treating technicalities with the constructions in singular homology. You may write down a list of simple identities relating e.g. compositions of these simple maps, and the analogous identities obtained by counter-functoriality on the $^*$-maps.

For any $n\in\mathbb{N}$ and for any $i\in[n]:=\{1,\dots,n\}$, you may consider the maps $\tau_i:[n-1] \to [n]$ defined in Iverson notation by $$\tau_i(x):=x+[x\ge i]\, .$$ That is, $\tau_i(x)=x$ unless $x\ge i$, in which case it is $x+1$). It induces by composition the map $\tau_i^*:\mathbb{N}^n\to\mathbb{N}^{n-1}$ that takes the element $x\in \mathbb{N}^n$ to $x\circ\tau_i$, which is what you want. In case of need, to recall the domain we may write $\tau_{i,n}$ instead of $\tau_i$; also, to simplify the notation, $\tau_i\cdot x$ instead of $\tau_i^*(x)$.

This and similar notations are somehow useful e.g. in treating technicalities with the constructions in singular homology. You may write down a list of simple identities relating e.g. compositions of these simple maps, and the analogous identities obtained by counter-functoriality on the $^*$-maps.