= Solution
Read the supplied <tangle> from top to bottom. Its input word is $\downarrow\uparrow\downarrow$, and its output word is $\downarrow$. There are three successive events: the upper crossing of the rightmost two strands, the lower crossing of the leftmost two strands, and the right-hand oriented evaluation. At both crossings, the upper-left-to-lower-right strand passes over; both are therefore $X_+$ in the geometric convention of part (a). The evaluation joins a downward strand to an upward strand, so it is $\overleftarrow{\cup}$.
The elementary factorization is consequently
$$
\boxed{T=(\downarrow\otimes\overleftarrow{\cup})
\circ(X_+\otimes\uparrow)
\circ(\downarrow\otimes X_+).}
$$
Its boundary words give an explicit composition check:
$$
\downarrow\uparrow\downarrow
\longrightarrow\downarrow\downarrow\uparrow
\longrightarrow\downarrow\downarrow\uparrow
\longrightarrow\downarrow.
$$
The third panel in part (b) draws these slices. Sliding the right-hand evaluation downwards and straightening the identity portions gives the PDF diagram without changing either crossing's overpass. If the names $X_+,X_-$ are interchanged in another crossing convention, both displayed crossings must be renamed together; the stated overpasses determine the actual <tangle> independently of those names.
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