I'm currently reading a review article called Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance by Jessica Striker. In this article, Striker writes that there is a ''nice'' bijection between the set of noncrossing matchings of $2b$ points and $SYT(2 \times b)$, the set of standard Young tableaux of size $2 \times b$. Here are two examples of the bijection:
Unfortunately, I cannot seem to understand how exactly this bijection works.
As can be seen in the figure, given any pair $(i, j)$ with $i<j$ in the noncrossing matching, we want to map this to this to a tableau with $i$ on the first row and $j$ on the second row. But it is neither clear to me why this map should be injective nor surjective.