Associativity isomorphism

A category, ${\displaystyle {\mathcal {C}}}$, is called semi-groupal if it comes equipped with a functor ${\displaystyle {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}}$ such that the pair ${\displaystyle (A,B)\mapsto A\otimes B}$ for ${\displaystyle A,B\in {\text{ob}}({\mathcal {C}})}$, as well as a collection of natural isomorphisms known as the associativity isomorphisms (or "associators").^{[1][2][full citation needed]} These isomorphisms, ${\displaystyle a_{X,Y,Z}:X\otimes (Y\otimes Z)\to (X\otimes Y)\otimes Z}$, are such that the following "pentagon identity" diagram commutes.