So some time ago when Mutiny was still working I experimented quite a bit with its fedimint functionality. I found that the same lightning invoice generated in the fedimint can be paid both from another member of the fedimint and from outside the fedimint (as expected).

Since payment withing the fedimint involves exchange and redemption of ecash tokens, how exactly the lightning invoice fits into this? Does the sender somehow determine that the invoice is from another member of the same federation? How does this work?