What Is a Morphism?

First, you're making too many assumptions. Why should morphisms be relations? Why should they have a graph? Why should A and B have elements?

Second, even with these assumptions, morphisms still don't need to be functions. Consider the category of Sets with arrows reversed. Then each morphism is like the reverse of a function. In particular, there are 2 distinct morphisms from {x,y} to {x} even though there is only one such function.