In Lurie's On the Classification of Topological Field Theories, he states in Proposition 1.1.8 that for an oriented compact manifold $M$ and a TQFT $Z:\mathrm{Cob}(n)\to \mathrm{Vect}_k$, there is a perfect pairing $Z(\overline{M})\otimes Z(M)\to k$ where $\overline{M}$ denotes $M$ with the opposite orientation. In the proof of this, he mentions the easily defined map $\alpha:Z(\overline{M})\to Z(M)^\vee$, where $^\vee$ denotes the dual space, which relies on the evaluation map coming from the cobordism $M\coprod\overline{M}\to\emptyset$ associated to $M\times[0,1]$. He intends to describe its inverse $\beta:Z(M)^\vee\to Z(M)^\vee\otimes Z(M)\otimes Z(\overline{M})\to Z(\overline{M})$ as a composite of the coevaluation map from $\emptyset\to M\coprod\overline{M}$ (i.e. $k\to Z(M)\otimes Z(\overline{M})$) followed by the bilinear pairing of $Z(M)$ with its dual. He then says that by "judiciously applying the axioms for a topological field theory, one can deduce that $\beta$ is an inverse to $\alpha$."
I am having trouble seeing that this is so. Just looking for maybe a little help, not necessarily the entire proof here, but anything would be appreciated? Is it futile to look at elements of the vector spaces considering that we don't really know much about what's going on there? Is this an entirely categorical proof, and if so, which "axioms" should I be looking the most closely at, the coherence ones for tensor functors?
Thanks!