$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark In the setting of Definition, we will refer to $(P)$ as the pentagon identity. It is a prototypical example of a coherence condition: the associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ “witness” the requirement that the tensor product is associative up to isomorphism, and the pentagon identity is a sort of “higher order” associative law required of the witnesses themselves.