Natural isomorphism of a monoidal category The definition of a Monoidal Category from "Categories for the working mathematician" says that it is a category equipped with tensor products, associative up to a natural isomorphism.  
What does "associative up to a natural isomorphism" mean in this scenario? I read the wikipedia definition of "up to" but cannot understand it when we are talking about categories.
I would appreciate any help.

Let's break it down.
A natural isomorphism is a natural transformation that is also an isomorphism. Or alternatively, a isomorphism (technically a family of isomorphisms) that is natural.
Associative "up-to natural isomorphism" means, somewhat informally, that $(A\otimes B)\otimes C$ is not necessarily equal to $A\otimes (B\otimes C)$, but there is an isomorphism $\alpha:(A\otimes B)\otimes C\to A\otimes (B\otimes C)$ which is natural.
