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.

alwaysconsists of rewriting of brackets, essentially by definition. How about a 2-group constructed from a group cohomology class; would you regard its associator as always "trivial"? $\endgroup$