What are some "natural" motivating examples of the following:


i) A strict monoidal category, 


ii) A monoidal with non-trivial associatots?


For i) the only examples I know are categories which have been strictified, are there any examples occuring "in nature" which are strict, or is strictness in some sense an "unnatural" or artificial requirement?


For ii) I should clarify what I mean by "non-trivial" - basically the examples I consider trivial are tensor products 
of vector spaces, bimodules, representations, and so on, where the associator is just the elementary rewritting of brackets.



Examples of strict monoidal categories and monoidal categories with nontrivial associators
i) [A monoid](https://en.wikipedia.org/wiki/Monoid)


ii) Representations of [a quasi-Hopf algebra](https://en.wikipedia.org/wiki/Quasi-Hopf_algebra)


