A map of locales $f : X \rightarrow Y$ is closed if it satisfies the reciprocity relation $f_*(f^*(X) \vee Y) \cong X \vee f_* (Y)$.

How can we express a that a map of schemes $f : X \rightarrow Y$ is closed in terms of the direct image $f^*$ and inverse image $f_*$? Is this equivalent to the same reciprocity relation?

Is there some tweaking of the setting for locales which gives rise to a similar reciprocity condition for schemes?

Any insights are appreciated.