Given adjoint functors $F: A \to B$, $G: B \to A$, if you then take their pullback functors $F^* : Set^B \to Set^A$ and $G^* : Set^A \to Set^B$ given by precomposition, are these two also adjoint (assuming conditions for their adjoints to exist)? According to wikipedia http://en.wikipedia.org/wiki/Functor_category, this is true for postcomposition, is it true for precomposition? I realise these functors have Kan extensions as adjoints but can find nothing in the literature about taking the Kan extension along the adjoint of a functor.
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This is best understood via the bicategorical characterization of adjunctions in terms of unit and counit 2morphisms. It is then clear that any bifunctor preserves adjunctions in the obvious sense. And taking presheaf categories and pullback functors is clearly a bifunctor from (small categories)$^{op}$ to categories. 

