Let $f: X \longrightarrow Y$ be a morphism of schemes. We have that the functors $$ f^{-1}: \text{Sh}(Y) \longrightarrow \text{Sh}(X) $$ and $$ f_{*}: \text{Sh}(X) \longrightarrow \text{Sh}(Y) $$ are an adjoint pair $ f^{-1} \dashv f_{*}$. In the case that $f$ is an open immersion, we further have an adjunction $$ f_{!} \dashv f^{-1} \dashv f_{*} $$ where $f_{!}$ is determined by extending by zero. Similarly, $f$ is a closed immersion, we have an adjunction $$ f^{-1} \dashv f_{*} \dashv f^{!} $$
Now it is my understanding at least (correct me if I am wrong), that we also have this right adjoint in the case that $f$ is proper, which generalizes the case of $f$ being a closed immersion. I was wondering if there is a similar generalization for the left adjoint of $f^{-1}$. Is there some way of generalizing "open immersion" in the same way that proper generalizes "closed immersion"?
If not, can it be done if we relax ourselves to presheaves? What about if we relax ourselves to locally ringed spaces or just ringed spaces?