ct.category theory - Must the left and right unitors of a monoidal category coincide at the neutral object?

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 $Iotimes I to I$ that can be constructed only from the unitors and the associator. (There may be plenty of other isomorphisms $Iotimes I to I$ 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.