Given that they are among the most commonplace concepts of category theory, monoidal categories are strangely unsatisfactory in some ways. In (weak) monoidal categories like those of sets or vector spaces, the associativity and unitality laws of a monoid hold only up to natural isomorphism, and the “associators” and “unitors” giving the natural isomorphisms need to obey certain coherence axioms. All that seems to be inevitable, but the particular selection of coherence axioms, known as Mac Lane’s pentagon and triangle identites, are rather mysterious at first sight. Why these axioms and not others?1
This is a companion discussion topic for the original entry at https://topos.site/blog/2023-08-15-unbiased-pseudomonoids/