Given any functor between two small categories, one may construct two functors between their presheaf categories. Let $f:\mathcal{A}\rightarrow\mathcal{B}$ be such a functor. Then we may extend this functor to a functor, $f_*\hat{\mathcal{A}\rightarrow\hat{\mathcal{B}$ by co-continuity. On the other hand wehave a functor, $f^{*}:\hat{\mathcal{B}}\rightarrow\hat{\mathcal{A}}$ defined by the composite, $X(f)$ My question is: are these two functors adjoint?