A definition of "adjoint" of a planar tangle is given by Jones/Kodiyalam as follows. Consider the tangle $T$
where the red dots are the distinguished points (aka first points) of the respective box. (If you are fit to answer this question you probably also know that the strings have orientation.)
The adjoint is now obtained by reflecting the tangle vertically, thereby reversing the orientation. We don't want that, so we simply reverse the orientation again. The new distinguished point for a box is now the point that was the last (before reflecting), in a clockwise sense. So for the tangle in question we get the adjoint $T^*$ as seen in the following picture:
My question is now: In what sense is this an adjoint? Given some planar algebra $P$, the first tangle gives rise to a linear map $Z_T:P_2\otimes P_6\rightarrow P_6$, so I would expect the adjoint to be a map $Z_{T^*}:P_6\rightarrow P_2\otimes P_6$, i.e. something that's going in the opposite direction.
The notion of adjoint given here seems like interpreting the tangle as going from top to bottom (or the other way around), while it actually has to be interpreted as "going from the inside to the outside" (meaning taking the adjoint would become some sort of "turning inside out")
Can anyone help me understand this definition, maybe provide a justification?
P.S.: I didn't find a fitting tag, unfortunately.