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 $O_Y$?

A ringed space is a pair $(X, \mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf of rings on $X$.
Then a morphism of ringed spaces from $(X,\mathcal{O}_{x})$ to $(Y,\mathcal{O}_{Y})$ is a pair $(f, \phi)$ where $f \colon X \to Y$ is a morphism of topological spaces and $\phi \colon \mathcal{O}_{Y} \to f_{*}\mathcal{O}_{X}$ is a morphism of sheaves (of rings on $X$).
Consider a module $\mathcal{F} \in \textrm{Mod}(\mathcal{O}_{X})$, then $f_{*}(\mathcal{F})$ is a sheaf on $Y$ and an $f_{*}\mathcal{O}_{X}$-module (which is also an $\mathcal{O}_{Y}$-module), so the push forward in this case is a functor from $\mathcal{O}_{X}$-modules to $\mathcal{O}_{Y}$ modules.
This is the direct image functor, which may be more conveniently written as a functor of abelian categories: $$f_{*} \, \colon \, \textrm{Mod}(\mathcal{O}_{X}) \to \textrm{Mod}(\mathcal{O}_{Y})$$
