Is there a name for this "homomorphism" between vector spaces over different fields? Let $V,W$ be vector spaces over fields $F_V,F_W$ respectively. If $F_V=F_W=F$ we have that a function $h:V\to W$ is a homomorphism from $V$ to $W$ if $h(v+w)=h(v)+h(w)$ and $h(\alpha v)=\alpha h(v)$, for all $v,w\in V$ and $\alpha\in F$.
But what if $F_V\neq F_W$? Then a "homomorphism" from $V$ to $W$ should be defined as two functions $h:V\to W$ and $g:F_V\to F_W$ such that $h$ and $g$ are homomorphisms and the following holds for all $v,w\in V$ and $\alpha\in F_V$:
$$h(v+w)=h(v)+h(w)$$
and
$$h(\alpha v)=g(\alpha)h(v)$$
Is this a valid "homomorphism" from $V$ to $W$, or two vector spaces over different fields cannot have the same structure?
 A: Similar questions have been asked several times before: 


*

*Why the morphisms of vector spaces, over different fields is not interesting?

*''Linear'' transformations between vector spaces over different fields .

*First Order Language for vector spaces over fields

*Mapping vector spaces over two different fields?
You'll find good references in all of these links. In particular, the notion of homomorphism you write down is (1) the natural notion of homomorphism between models of the two-sorted theory of vector spaces over an arbitrary field (in the sense of model theory),  (2) closely related to the notion of a semi-linear map between vector spaces, (3) an example of a fibered category construction. 
Let me also point out that any such morphism can be factored into two stages. First, pick a field extension $g\colon F_V\to F_W$ (as Matthew Leingang points out in the comments, every homomorphism of fields is injective). This gives $F_W$ the structure of an $F_V$-algebra, and we can form the tensor product $V' = (F_W\otimes_{F_V} V)$. This is the "extension of scalars" which turns $V$ into an $F_W$-vector space. Finally, pick an $F_W$-linear map $h'\colon V'\to W$.  
It turns out that the pairs $(g,h')$, where $g\colon F_V\to F_W$ is a field extension and $h'\colon V'\to W$ is an $F_W$-linear map, are in natural bijection with the morphisms $(g,h)$ in the sense of your question. 
