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?

Note: here $f_*$ and $f^*$ are the direct image and inverse image functors between categories of quasicoherent sheaves on $X$ and $Y$.

Any insights are appreciated.

The map $f^*((f_* M) \otimes N) \rightarrow M \otimes f_* (N)$ arises as the adjunct of the map $$ f^* ((f_* M ) \otimes N) \rightarrow (f^* f_* M) \otimes (f^* N) \rightarrow M \otimes (f^* N)$$