Is the unit of an adjunction an epimorphism?

Question

I was wondering, does the unit of an adjunction of functors need to be an epimorphism in general?

Answer 1

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 $S$.

Answer 2

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.