Consider the two statements:


1. "Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra", as stated in [1506.03546](https://arxiv.org/abs/1506.03546) page 4. The above paper refers to (I think) Theorem 7.6 of this [paper](https://core.ac.uk/download/pdf/82755356.pdf). In the next paragraph of 1506.03546, they say that this is true for non-unitary fusion categories as well.
2. Not every unitary fusion category is strict, e.g. the unitary fusion category associated to any finite group G and a nontrivial 3-cocycle of G.


Now, I was previously under the impression that any category of endomorphisms is strict, but the two statements above show that this is wrong. What is a simple example that illustrates that a category of endomorphisms can have nontrivial associator?



Realizing a fusion category as endomorphisms of an algebra
Every monoidal category is monoidally equivalent to a strict one. This is MacLane’s famous coherence theorem.


What the example $\mathrm{Vec}(G, \omega)$ shows is that you can’t always make it simultaneously strict and skeletal. But you can do either separately if you like.


