Is antipode unique for bialgebras in arbitrary monoidal categories? If $B$ is a bialgebra in the category $\tt{Vect}$ of vector spaces (over $\mathbb C$, for example) then $B$ can't have two different antipodes.
Is this true for bialgebras in an arbitrary symmetric monoidal category?
 A: Nice answer by Evan Jenkins. (Here are references for string diagrams: http://ncatlab.org/nlab/show/string+diagram#details)
The bottom line is that being a Hopf algebra is a property of a bialgebra rather than an extra structure. A Hopf algebra is usaully defined as a bialgebra with an antipode. Then the "propertiness" means that the antipode is unique. This indeed holds in any braided monoidal category. 
However, there is another definition of a Hopf algebra which emphasizes the "propertiness". It is based on the generalization of the following simple group-theoretic fact: A monoid is a group if and only if the maps 
$$(a, b) \mapsto (ab, b), \qquad (a, b) \mapsto (a, ab)$$
are invertbile. Indeed, the inverses are given by
$$(a, b) \mapsto (ab^{-1}, b), \qquad (a, b) \mapsto (a, a^{-1}b).$$
Generalizing this to bialgebras (within any braided monoidal category), a Hopf algebra is a bialgebra $B$ for which the maps
$$B\otimes B \xrightarrow{1\otimes d} B\otimes B\otimes B \xrightarrow{m \otimes 1} B\otimes B$$
$$B\otimes B \xrightarrow{d\otimes 1} B\otimes B\otimes B \xrightarrow{1\otimes m} B\otimes B$$
are invertible. These maps are called left and right fusion maps.
This definition is equivalent to the definition via an antipode. The anipode can be recovered as
$$B \xrightarrow{1\otimes \eta} B\otimes B \xrightarrow{l^{-1}} B\otimes B \xrightarrow{\epsilon\otimes 1} B$$
where $l$ is the left fusion map. Or it can be recovered by a similar composite involving the inverse of the right fusion map.
A: Yes, as the following string diagram proof shows.

