I've recently started reading about sheafs and ringed spaces (at the moment, primarily on wikipedia). Assuming I'm correctly understanding the definitions of the direct image functor and of morphisms of ringed spaces, a morphism from a ringed space $(X, O_X)$ to a ringed space $(Y, O_Y)$ is a continuous map $f\colon X\to Y$ along with a natural transformation $\varphi$ from $O_Y$ to $f_*O_X$.
Why does the definition require $\varphi$ to go from $O_Y$ to $f_*O_X$ as opposed to from $f_*O_X$ to $O_Y$?