TITLE: Examples of "Monoidal Categories" without Pentagon Axiom
QUESTION [5 upvotes]: One of the main goals of asserting the pentagon axiom for monoidal categories is to get the coherence theorem. I wondered how a typical example would look like that does not satisfy the theorem.
More concretely, I'm looking for a nice enough example of a "monoidal category" that has all the data of a monoidal category (as in e.g. nLab) but does not satisfy the pentagon axiom. (I'd already be happy for an example that satisfies neither the pentagon nor the triangle axiom.)
One could probably just write down a small example by hand, though I'm not sure how typical such an example would be. I also found arxiv:0109088 but I'm admittingly a bit lazy to go through the author's example $\mathbf{IRBTree}$ which also looks a bit too complicated for my purposes.

REPLY [6 votes]: I asked this question on the German forum MP/249861 and got a nice answer there.
Let e.g. $(\mathscr{C}, \otimes, \mathbf{1}, \lambda, \rho, \alpha)$ be a monoidal pre-additive category with left/right unitor $\lambda, \rho$ and associativity constraint $\alpha$. Then, $(\mathscr{C}, \otimes, \mathbf{1}, -\lambda, \rho, - \alpha)$ satisfies everything but the pentagon axiom.