Checking when a linear isomorphism is natural I am thinking about what it means for linear maps to be "natural" and in particular I am thinking about the following example.  Let $V$ be a finite-dimensional real vector space with dimension $n$.  Then Hom$_{\mathbb{R}}(V\otimes\mathbb{C},\mathbb{R})\cong$ Hom$_{\mathbb{R}}(V,\mathbb{R})\otimes\mathbb{C}$ since they both have real dimension $2n$.  My question is whether or not there is a definition for this isomorphism that is independent of choice of basis and why.  It seems to me that this cannot be the case since we'd like to define the image of any $\mathbb{R}$-linear map $f$ on $V\otimes\mathbb{C}$ in Hom$_{\mathbb{R}}(V,\mathbb{R})\otimes\mathbb{C}$ as $f|_{V}\otimes \lambda$ where $\lambda$ is either $1$ or $i$, but how would you know which to choose?  How can I prove this more rigorously?
Edit: I am comfortable with the meaning of "natural" and "independent of choice of basis".  At this time I would just like to know whether or not such a natural isomorphism exists.
 A: The idea that a map is 'natural' is sometimes informally stated as being independent of choice of basis, but what it really means is that it's a component of a natural transformation between two functors.
Here, the functors are $\operatorname{Hom}(-\otimes \mathbb{C},\mathbb{R})$ and $\operatorname{Hom}(-,\mathbb{R})\otimes\mathbb{C}$.  The question of whether they're naturally isomorphic requires you to produce a natural isomorphism between these two functors, which means a choice of isomorphism $$\alpha_V:\operatorname{Hom}(V\otimes \mathbb{C},\mathbb{R})\to\operatorname{Hom}(V,\mathbb{R})\otimes\mathbb{C}$$ for every choice of vector space $V$ such that for any map $f:W\to V$, you obtain a commutative square
$$\begin{matrix}&\alpha_W&\\\operatorname{Hom}(W\otimes \mathbb{C},\mathbb{R})&\cong&\operatorname{Hom}(W,\mathbb{R})\otimes\mathbb{C}\\\downarrow &&\downarrow\\\operatorname{Hom}(V\otimes \mathbb{C},\mathbb{R})&\cong&\operatorname{Hom}(V,\mathbb{R})\otimes \mathbb{C}\\&\alpha_V&
 \end{matrix}$$
These two functors may be naturally isomorphic, but, at least to me, not in an obvious way.
