 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 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.