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. 




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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?



Finite categories and partial orders
Almost, this has nothing to do with finiteness: any category where the
homsets have at most one element each is a preorder. Define
$a\le b$ if there is an arrow from $a$ to $b$. Then $\le$ is reflexive
and transitive, by the category axioms. But it may fail to be
antisymmetric: one may have $a\le b\le a\ne b$. But a preorder
induces a partial order on the equivalence classes with respect to
the relation $a\sim b$ if $a\le b\le a$: in the category these
are isomorphism classes of elements.


