TITLE: When is a category isomorphic to its opposite?
QUESTION [6 upvotes]: I could verify that if Morph$(A, B)$ is in bijective correspondence with Morph$(B, A)$ for all objects $A, B$ in a category then one shall construct isomorphism between that category and it's opposite.  Is it a necessary condition.  Seems like it is. But I'm not really convinced of my justification.

REPLY [12 votes]: Your condition is neither necessary nor sufficient.
To see that it's not necessary, consider the category with two objects $A$ and $B$, and one morphism from $A$ to $B$. In this category, $|\hom(A,B)|=1$ and $|\hom(B,A)|=0$, so it doesn't satisfy your condition. However, it is isomorphic to its opposite — it's just that the isomorphism exchanges $A$ and $B$.
To see that it's not sufficient, note that not all monoids are isomorphic to their opposites. But, as a one-object category, any monoid will trivially satisfy your condition.