Well, I'm afraid that people will believe you! For your purposes it might not matter whether all bijections are isomorphisms, but it does matter to other peopl

Well, I'm afraid that people will believe you! For your purposes it might not matter whether all bijections are isomorphisms, but it does matter to other people (for practical as well as theoretical purposes). What worries me is that someone may come along, see a nicely-written article appearing to summarise the nature of some important classes of functions, take it at face value, and go away believing that all bijections are isomorphisms, only to have this belief trip them up at some later stage. Not everyone is going to hop on Wikipedia or open up a maths textbook to verify your assertions, after all.