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 $O_Y$?
Best Answer
1) The model one has to keep in mind is that $X$ and $Y$ are geometric spaces equipped with functions of a certain degree of regularity defined on their open subsets.
A typical example would be that $X$ and $Y$ are $C^\infty$ manifolds, that for $V\subset Y$ one takes $\mathcal O_Y(V)=C^\infty (V)$ and similarly for $X$.
If $f:X\to Y$ is a $C^\infty$ map one gets for each open subset $V\subset Y$ a composition map $f^{*}_V : C^\infty (V)\to C^\infty (f^{-1}V):g\mapsto g\circ f$ and letting $V$ vary the $f^{*}_V$ yield a morphism of sheaves of rings on $Y$:$$\mathcal O_Y\to f_*\mathcal O_X$$ Do you see that there is no way the arrow could go in the other direction?
2) For general ringed spaces the $\mathcal O_Y(V) $ 's are no longer required to be functions but are abstract rings and the $f^{*}_V : \mathcal O_Y (V)\to \mathcal O_X (f^{-1}V)$ are no longer compositions but are given as part of the structure: they are exactly the datum of a morphism of sheaves $\mathcal O_Y\to f_*\mathcal O_X$ .
Beware that a morphism of ringed spaces is a very general concept :
Any morphism of rings $A\to B$ can be seen as a morphism of one-point ringed spaces $(\lbrace * \rbrace ,B)\to (\lbrace * \rbrace,A)$ .
But usually one only studies locally ringed spaces, a non-full subcategory of the category of all ringed spaces, the best known example being the category of schemes.