Iterating a Binary Operation along a Finite List

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\ast$ be a binary operation on a set $X$, let $n\in\mathbb{N}$, let $[n]$ be as in Intervals of Natural Numbers §segment, and let $a:[n]\to X$, $k\mapsto a_{k}$.

¶ There is exactly one map $s:[n]\to X$ such that $s(1)=a_{1}$ and $s(k+1)=s(k)\ast a_{k+1}$ for every $k\in[n]$ with $k<n$.