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).

Categories for the Working Mathematicianas one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual pentagon and triangle diagrams) that define a monoidal category. $\endgroup$On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc.(Journal of Algebra 1, 397–402) $\endgroup$2more comments