Does the equality of map of stalks imply the equality of the map of global sections?

Question

Let $X$ be a scheme and $R$ be a commutative ring with unity. Let $f,g: X \to \operatorname{Spec}{R}$ be morphisms of schemes. Suppose I know that $f(x) = g(x)$ and $f^{\#}_x = g^{\#}_x$ for all $x \in X$, where $$ f^{\#}_x, g^{\#}_x: R_P/(P R_P) \to O_{X, x}/m_x $$ the induced map on the residue fields. Does is then follow that $f^{\#} = g^{\#}$? where $f^{\#}$ and $g^{\#}$ are maps of the global sections $$ R \to \Gamma(X, O_X). $$ Thank you!