A bijection is an injective and surjective function, as such:
But the following is also a bijective function:
This means there is a further restriction that could be added to the definition of a function, and that is one where "no crisscrossing of lines is allowed." This is what I mean when I say "a bijection that must preserve the order of its mappings."
Is there a category or name for such a restricted relationships between sets and/or categories?
It is called an increasing (or less ambiguously: order-preserving) bijection. Note that it is not necessarily an isomorphism of ordered sets. It is however if its domain is totally ordered.