I'm trying to read through this paper. Below is a quote from it found on page 9.
Indeed, $u(1)$ is the unit of the multiplication map $\mu$ since the corresponding composition of bordisms is diffeomorphic to the identity map on $S^1$
What exactly are the "corresponding composition of bordisms" in this case?
The bordism $\mu \circ (u\otimes 1)$ results as the juxtaposition of the "cap" represented by $u$ and an identity map, composed with $\mu$: this is what you want to commute, and why it commutes in $\bf Bord$: