Remark 2.1.1.6. In the setting of Definition 2.1.1.5, 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.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$