Consider the two statements:
"Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra", as stated in 1506.03546 page 4. The above paper refers to (I think) Theorem 7.6 of this paper. In the next paragraph of 1506.03546, they say that this is true for non-unitary fusion categories as well.
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?