Two different definitions of sheaf of $K$-modules and tensor products. I am confused by two different approaches to defining sheafs of modules.  In Hartshorne there is the concept of a sheaf $F$ of modules $O_X$-modules, where $F(U)$ is a module over $O_X(U)$ with compatibility of restrictions.  
In Frank Warner's book Differential Geometry and Lie Groups, sheaves of $K$-modules are defined for a ring $K$ to just be a sheaf such that each section is a $K$-module and restrictions are $K$-linear.  Presumably such sheaves are then sheaves over the constant sheaf $K$ in a natural way.  
However, I'm confused by conflicting definitions of tensor products.  In Hartshrone the tensor product of two sheaves of $O_X$-modules, $F$ and $G$, is defined as the sheaf associated to the presheaf $U\mapsto F(U)\otimes_{O_X(U)} G(U)$.
In Warner the tensor product of two sheaves of $K$-modules, $F$ and $G$, is defined as the sheaf associated to the presheaf $U\mapsto F(U)\otimes_K G(U)$.  
What happens when we consider the tensor of two sheaves of $K$-modules $F$ and $G$ as in Warner's definition as a sheaf of modules over the sheaf $K$?  Is it the same as if we first considered $F$ and $G$ as sheaves of modules over the sheaf $K$ and then took the tensor product as in Hartshorne's definition?  The complication is of course that $K(U)$ need not equal $K$.  
Edit: Actually I'm thinking that there is a natural map $F(U)\otimes_K G(U) \to F(U)\otimes_{K(U)} G(U)$ that gives a morphism of sheaves which is an isomorphism on stalks since the stalk of both sides at $x$ is $F_x\otimes_K G_x$.  
 A: So we have a ringed space $(X,\mathcal K)$ where $\mathcal K$ denotes the constant sheaf associated to the commutative ring $K$ (all stalks of this sheaf are isomorphic to $K$) and we have $\mathcal K$-Modules $F$ and $G$.
For every $U\subseteq X$ open, we have a map $K\to \mathcal K(U)$ (sending $k\in K$ to the constant section above $U$ with value $k$) and we have a (surjective) natural map 
$$\phi_U\colon F(U)\otimes_K G(U)\to F(U)\otimes_{\mathcal K(U)} G(U)$$ 
of $K$-modules; it is the one induced by the $K$-bilinear map $F(U)\times G(U)\to F(U)\otimes_{\mathcal K(U)} G(U)$ given by $(x,y)\mapsto x\otimes y$. These maps are compatible with restriction of $U$ so applying sheafification yields a morphism of $K$-Modules $\phi\colon F\otimes_{\text{Warner}} G\to F\otimes_{\text{Harts}} G$. We want to know which maps it induces on the stalks. By taking the direct limit over the inductive system of neighbourhoods of a point $x\in X$ (and using the fact that tensor products commute with direct limits) we find the isomorphism you mentioned:
$$F_x\otimes_K F_x\to F_x \otimes_{\mathcal K_x} G_x = F_x \otimes_K G_x.$$
What do you think about it?
