I was wondering: is the unit of an adjunction an epimorphism?

In general.

-
I can hardly think of any cases where the unit of an adjunction is an epimorphism. Perhaps you meant to ask whether the counit of an adjunction is an epimorphism? –  Uday Reddy Feb 2 '13 at 10:45

The unit of said adjunction is the map that sends a set $S$ to the (underlying set of the) vector space whose basis is $S$.
The answer is no. Consider the category of sets $\mathsf{Set}$ and the category $\mathsf{Set}/I$ for some set $I$ with more then one element. There is an adjunction between these categories which consist of the constant family functor $\Delta: \mathsf{Set}\to \mathsf{Set}/I$ and the $I$-indexed product functor $\prod: \mathsf{Set}/I \to \mathsf{Set}$. The unit of this adjunction is almost never an epimorphism, which here is just a surjective function.