= Solution
Horizontal juxtaposition is the <tensor product> in the <tangle category>. The first panel gives a representative of the first expression: from left to right it consists of the upward identity, the ordinary <evaluation morphism>, the downward identity, the reversed <coevaluation morphism>, and the negative crossing. The two arcs are disjoint from the other strands and the crossing is the only over/under intersection in that panel.
The second expression has a boundary-count defect \b[as printed]. Under the top-to-bottom convention fixed in part (a), $X_+\otimes\uparrow\otimes\uparrow$ has four input and four output endpoints, whereas $\downarrow\otimes\cap\otimes\uparrow$ has only two input and four output endpoints. Those successive boundary words cannot be glued. Reading bottom-to-top does not fix it: the middle factor then has four inputs and two outputs, but the final $\cup$ has no inputs. Thus there is no representative of that literal composite in the <tangle category>.
An explicit composable correction replaces the middle arch by an <evaluation morphism> and orients both evaluations to match the strands. Write $e_{\downarrow\uparrow}=\overleftarrow{\cup}$. One such corrected morphism is
$$
\boxed{e_{\downarrow\uparrow}\circ
(\downarrow\otimes e_{\downarrow\uparrow}\otimes\uparrow)
\circ(X_+\otimes\uparrow\otimes\uparrow).}
$$
Its input word is $\downarrow\downarrow\uparrow\uparrow$. The crossing exchanges the first two downward strands, the middle evaluation removes strands two and three, and the outer evaluation removes the remaining downward/upward pair. The second panel is its original drawing. It has four top endpoints and no bottom endpoints. This is a qualified correction, not a claim that the ill-typed printed composite defines a morphism.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-2-tangles.png]
{title=Oriented tangle representatives, a composable correction of the printed composite, and elementary slices of the supplied tangle}
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