Morphisms between Locally Ringed Spaces

Locally ringed spaces are generalization of differential manifolds, and classical algebraic varieties. The notion of morphisms of ringed spaces then, should be a generalization of the notion morphisms in the latter categories. Thus, to define morphism between ringed spaces, we need to first be able to rephrase the notion of differentiable maps and morphisms of varieties in a way that only uses the structure sheaf.  (Recall that classically, varieties are irreducible subset of $k^n$ with Zarisky topology, and their morphisms are defined to be restrictions of polynomial maps). The idea is that a continuous map between two differentiable manifolds is differentiable iff it pullbacks differentiable functions to differentiable functions. In the case of classical varieties, a morphism is a continuous map which pullbacks regular functions to regular functions.

In a general ringed space $X$, the sections of $O_X$ cannot be canonically interpreted as functions, so a morphism of ringed spaces $(X, O_X)$ and $(Y, O_Y)$ consists of a pair $(f, f^{\sharp})$ where
$f: X \to Y$ is continuous
and $f^{\sharp}: O_Y \to f_* O_X$.
This $f^{\sharp}$ is an artificial version of pullback of $f$.

On the other hand, if $X$ is a locally ringed space, then given a section $s \in O_X(U)$ and $x \in X$, we can define $s(p)$ to be the image of $s$ in $O_X(U) \to (O_X)_x \to (O_X)_x/\mathfrak{m}_x = \kappa(x)$. Thus we can put restriction of $f^{\sharp}$ to make it resemble a pullback. We will require that

if $t$ vanishes $f(x)$ then  $f^{\sharp} (t)$ vanishes at $x$.

In other words, if $t \in \mathfrak{m}_{f(x)}$  then $f^{\sharp}(t)$ is in $\mathfrak{m}_x$, or more concisely,

$f^{\sharp}_x: (O_Y)_{f(x)} \to (O_X)_x$ is a local homomorphism.