# Is every functor a composition of adjoint functors?

My understanding of Ben's answer to this question 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?

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A similar but simpler argument: any adjunction between categories $C$ and $D$ induces a bijection between their sets (or classes) $\pi_0 C$ and $\pi_0 D$ of connected-components. So categories with different numbers of connected-components are never linked by a chain of adjoint functors. Eric's example will also do here. –  Tom Leinster Aug 16 '12 at 15:02