I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theory is related to purely functional programming languages). But, as the book is very succint, may be it lacks some information so I have a question.

Supose I have a category with a finite number of objects {a1, a2, ..., an}. Supose also that, if there is a morphism f : ai -> aj connecting two objects, then it is unique. Does this category always correspond to some partial order in the set {a1, a2, ..., an}?

I convinced myself drawing some diagrams that this could be the case (couldn't find a counter example), but I'm not sure.

edit: I see your point. I don't have antisymmetry garanteed.

Suppose I further restrict things: I'm not going to allow a morphism (b -> a) it there's already a (a -> b). Then I fix this, right? It'll be a partial order with no equalities, right?