$f\colon K\to L$ is a map of sets,and $f^*\colon \mathcal {P}(L)\to\mathcal{P}(K)$ is a functor sending a subset $S$ of $L$ to to its inverse image $f^{-1}(S)$. I need to find left and right adjoints to $f^*$.
Right adjoints: Functor $G\colon \mathcal{P}(K)\to\mathcal {P}(L)$ is a right adjoint to $f^*$ iff there exist natural transformations $\eta\colon 1_{\mathcal {P}(L)}\to Gf^*$ and $\epsilon:f^*G\to1_{\mathcal{P}(K)}$ satisfying triangle identities. This holds iff $S\subseteq G(f^{-1}(S))$ for every $S\subseteq L$ and $f^{-1}((G(T))\subseteq T$ for every $T\subseteq K$.
Am I correct? Can I say something more about right adjoints?