= Solution
The <tangle category> has objects finite words in the two orientations $\uparrow,\downarrow$, including the empty word. A morphism is an oriented compact one-dimensional manifold, embedded in a three-dimensional slab, whose boundary is the prescribed ordered set of top and bottom endpoints. Closed components are allowed. Two such embeddings represent the same morphism when they are ambient-isotopic relative to the boundary.
Read diagrams \b[from top to bottom] here, matching the maps drawn in part (d). Composition stacks diagrams with exactly matching intermediate boundary words; horizontal juxtaposition is the <tensor product>. The empty word is its unit, and straight oriented strands are identities. Crossings, the two oriented evaluations and the two oriented coevaluations generate all morphisms: place an embedding in generic position and slice between its crossings and local extrema. The <Reidemeister moves> and the cup/cap isotopies describe changes of diagrams without changing the underlying <tangle>. This is the ordinary unframed version; a framed <tangle category> instead retains framing data and has a modified first-move rule.
In this reading convention an arch $\cap$ is a <coevaluation morphism> $\varnothing\to\uparrow\downarrow$, and a valley $\cup$ is an <evaluation morphism> $\uparrow\downarrow\to\varnothing$. An overleft arrow reverses these strand orientations. We use $X_+$ for the crossing whose upper-left-to-lower-right strand passes over the other, and $X_-$ for its inverse; this geometric specification avoids ambiguity in sign conventions.
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