My understanding of Ben's answer to [this question](https://mathoverflow.net/questions/263/what-is-the-universal-property-of-associated-graded) is that even though associated graded is not an adjoint functor, it's not too bad because it is a composition of a right adjoint and a left adjoint.


But are such functors really "not that bad"? In particular, is it true that **any** functor be written as the composition of a right adjoint and a left adjoint?



Is every functor a composition of adjoint functors?
The answer is no, because the nerve functor turns an adjoint pair of functors between categories into inverse homotopy equivalences between spaces (this is because of the existence of the unit and counit and the fact that nerve turns natural transformations into homotopies). In particular, this means that any functor whose nerve is not a homotopy equivalence cannot be a composite of adjoints. For a very simple example, you could take the functor from the 2-object discrete category to the terminal category.


