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?

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, $f_{\ast}: \hat{\mathcal{A}} \rightarrow \hat{\mathcal{B}}$ by co-continuity. On the other hand we have a functor, $f^{\ast}:\hat{\mathcal{B}}\rightarrow\hat{\mathcal{A}}$ defined by the composite, i.e., by taking $X \mapsto X \circ f^{op}$. My question is: are these two functors adjoint?