Must the left and right unitors of a monoidal category coincide at the neutral object? A monoidal category has (among other things) a pair of natural transformations called the left and right unitors
$$ \lambda_A:I\otimes A \cong A $$
  $$ \rho_A:A\otimes I \cong A $$
The categories I work with on a daily basis have $\lambda_I=\rho_I:I\otimes I \cong I$, but I think that is mostly a consequence of the fact that most of them have a terminal object as their $I$.  Is this true in general?
I'm sort of hoping this is true, because otherwise it seems that some of the definitions of enriched categories need to make an arbitrary choice of one unitor or the other.  For example, see display (1.10) of Max Kelly's introduction to enriched category theory on page 10.
(I'm actually interested in the even-more-general case of premonoidal categories, but mathematicians don't use those very often, so I've posed the question in terms of monoidal categories instead).
 A: In Categories for the Working Mathematician MacLane included $\lambda_I = \rho_I$ as one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual pentagon and triangle diagrams) as the axioms defining monoidal categories.
It was however proven to follow from the other axioms in Max Kelly's 1964 paper On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. (Journal of Algebra 1, 397–402)
EDIT: For the case of premonoidal categories, I think $\lambda_I = \rho_I$ follows from the results of section 4 of John Power's Premonoidal categories as categories with algebraic structure (Theoretical Computer Science 278, 1-2, 303-321). In that paper, the unitors in a premonoidal category are defined to be central natural transformations; further down, the centre of a premonoidal category is defined to be the category with the same objects but with only the central morphisms of the original category, and this turns out to be a monoidal category, hence reducing to Kelly's proof.
A: The coherence conditions for a monoidal category assure that if an isomorphism of two expressions follows from the built-in natural isomorphisms, then it is the unique such isomorphism.  In particular, there is a unique isomorphism $I\otimes I \to I$ that can be constructed only from the unitors and the associator.  (There may be plenty of other isomorphisms $I\otimes I \to I$ in your particular monoidal category; there is a unique one in the free monoidal category.)
This is a general philosophy in n-category theory.  A collection of "coherence" axioms are "good" if they imply that the space of choices is contractible.  Recall that an $n$-category is contractible if it is nonempty, it is an $n$-groupoid, and for each object (it suffices to check at any particular object) the endomorphisms of that object are a contractible $(n-1)$-category.  I.e. a contractible $n$-category is one that's $n$-equivalent to $\{\rm pt\}$.  The theory of monoidal categories is an example of a theory with "good" coherence axioms: the space of isomorphisms that follow from the monoidal structure between any two objects is contractible.  Other examples of "good" theories include the usual theory of (only weakly associative) 2-categories, and Lurie's theory of $(\infty,1)$-categories.
Unfortunately, I don't know what a "premonoidal category" is, so I don't know if its coherence axioms are good in the above sense.
