EDIT: Fixed for clarity of intention
I was reading "The FFX Mode of Operation for Format-Preserving Encryption" when I came across Figure 1 (see below). It's obvious why it's reversible (top-down then down-top) but it's not clear why it should strictly be bijective instead of not-injective or not-surjective? What rules out the possibilities of collisions or skips that make it strictly one-one in both directions?
For example: The round function (Fk) can be block cipher or hash based and we pick a subset of output bits into the next round. A collision at say, Fk(n,T,0), in round zero would then map two inputs at the top to a single input at the bottom. Why is that not an issue? Sorry if I'm missing something but it's been a long night and have missed my morning coffee.
Example Feistel network from FFX paper: