definition of morphism of ringed spaces 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 $O_Y$?

Think about what it means to give a morphism from $\mathcal O_Y$ to $f_* \mathcal O_X$: it means that for every open set $V \subset Y$, there is a map 
$$\mathcal O_Y(V) \to \mathcal O_X\bigl( f^{-1}(V) \bigr).$$
If you imagine that $\mathcal O_X$ and $\mathcal O_Y$ are supposed to be some sorts of "sheaves of functions" on $X$ and $Y$, then this accords perfectly with the intuition that a morphism of ringed spaces should allow us to "pull back" functions.
Indeed, in concrete examples (such as smooth manifolds equipped with the structure sheaf of smooth functions), the map $\mathcal O_Y \to f_* \mathcal O_X$ is just the pull-back map on functions.
A morphism in the opposite direction doesn't have any analogous intuitive interpretation, and doesn't accord with what happens in the key motivating examples.
