Let $X\overset{f}{\to}Y$ be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by the limit formula for right Kan extensions $$U\mapsto \varprojlim G _{U\supset 𝑓^{−1}V}GV$$ where $G$ is a presheaf on $Y$ and the colimit is over opens $V\subset Y$ satisfying the condition in the formula.
On a formal level, sheafifying this adjoint does not give something obviously satisfying since sheafification is left adjoint, so there are no adjunctions to compose.
When life is good, the derived proper direct image admits a right adjoint which captures duality. Is it in any way related to the right adjoint of presheaf direct image?
Is the right adjoint defined above hiding in plain sight in some familiar constructions?
