In a category with one object and three morphisms id, a, and b, what is $a \circ b$?

Question

I am trying to understand basic category theory, and it seems to me like there are 3 possibilies: this composition could be any of the 3 morphisms (id, a, or b) without losing any properties of the category.

So are there several categories with one object and 3 morphisms? I feel as though there is only one, but then there is no concrete answer to $a \circ b$.

Thanks.