1. The problem statement, all variables and given/known data Show that there are only two possible categories with one object and two morphisms. 2. Relevant equations None 3. The attempt at a solution My thinking here is as follows : Let's say that the object in our category is A, and that the two morphisms are f and 1, where 1 is the identity. Obviously they both have domain = codomain = A. So the only way that I see to make a difference between the categories is to define the composition differently, i.e. to make either ff = f or ff = 1. Is this enough to say that the two are different?