I am reading a lecture note about tensor category by P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. The link is attached here https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/pages/lecture-notes/
In the definition 1.1.1（The definition of monoidal category), a family of isomorphism is given: $$ a_{X,Y,Z}: (X\otimes Y)\otimes Z\to X\otimes (Y\otimes Z)\quad X,Y,Z\in \mathcal{C} $$ $\otimes$ denotes a bi-functor: $\mathcal{C}\times \mathcal{C}\to \mathcal{C}$. The authors say that this is a "functorial isomorphism". What does it mean? I think $(X\otimes Y)\otimes Z$ and $X\otimes(Y\otimes Z)$ can be understood as tri-functors $\mathcal{C}\times \mathcal{C}\times \mathcal{C}\to \mathcal{C}$. And $a$ is understood as a series of natural isomorphisms between these two tri-functors. Is this understanding precise?
