How to identify "naturally" two subspaces?

Let \[ H = \{f \in (\mathbb F^n)^* \mid f(x_1, \ldots, x_n) = \sum c_ix_i \text{ and } \sum c_i = 0 \}. \] There's a map $(\mathbb F^n)^* \to W^*$ given by restriction—if you like, this is the dual $i^*$ of the inclusion $i\colon W \to \mathbb F^n$. Restrict this to a map $H \to W^*$ and show that you obtain an isomorphism. For this, you could use the fact that the kernel of $i^*$ is precisely $W^0$. What is $W^0 \cap H$?

I take the word "natural" here to mean, "Don't just identify $W$ and $H$ because they have the same dimension." The word often means "basis-free" (or functorial), but here it seems to me that we've chosen a basis by merely writing $\mathbb F^n$. In general, if I have a vector space $V$ and a subspace $W \subset V$ then the dual of $W$ is most naturally identified with the quotient $V^*/W^0$. (In the same spirit, can you find something naturally isomorphic to $(V/W)^*$?)