Definitions of ordinary functor categories and higher categories are considered with very similar algebraic and geometric methods such as graph structures and simplicial sets. I know the differences between ordinary algebra and higher algebra. But are there deeper relations or results between ordinary functor categories and higher categories to be characterized or analyzed ?



Relations between ordinary functor categories and higher categories
Maybe this doesn't address completly to the question, but I think it's a start.


Functor categories and higher categories are quite different objects,
the only relation that I can think of is that usually higher categories are defined either as objects in a presheaf-category satisfying a property (for instance as simplicial sets satisfying the horn-filler condition, in the case of qausi categories) or as algebras over a monad defined on a presheaf category (for instance a algebras for the initial operad with contraction in Batanin/Leinster's definition of higher category).
Also there's Gothendieck/Maltsiniotis' definition of higher category that models higher categories as *models for a sort of generalized algebraic theory* (at least that what I understand), that means that also in this case an higher category is a special object that lives in a presheaf category (i.e. a functor category). 


I don't know if there's anything else I can add, but I hope this could be a start.


