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

In general.

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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

A common example of an adjunction is the free functor and forgetful functor for a kind of universal algebra -- e.g. real vector spaces.

The unit of said adjunction is the map that sends a set $S$ to the (underlying set of the) vector space whose basis is $S$.

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and every space has a basis so that's epi. –  user58512 Jan 21 '13 at 19:52
Not every vector is a basis element, though! In fact, most aren't. –  Hurkyl Jan 21 '13 at 19:57
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.