There are a few relevant concepts here. First, let's just talk on the level of sets. Suppose $A, B, C$ are sets. Then fixing any function $f : A \times B \to C$, we can take an element $1 \to A$ of $A$ and compose it with $f$ to get a function $B \to C$, or in other words an element of $\text{Hom}(B, C)$. The category of sets is enriched over itself, so we can talk about homs internally, and the above defines a map
$\text{Hom}(A \times B, C) \mapsto \text{Hom}(A, \text{Hom}(B, C))$
which, as it turns out, is a natural isomorphism. This is the defining property of a Cartesian closed category and is, as Jim Belk says, also called currying.
Second, let's go to the level of finite-dimensional vector spaces (I think things get messed up in general). The correct replacement for a function $f : A \times B \to C$ is a bilinear function $f : A \times B \to C$, or equivalently a linear function $f : A \otimes B \to C$. The category $\text{FinVect}$ of finite-dimensional vector spaces is also enriched over itself, and we'd like to say that it also has the structure of a Cartesian closed category, except that we can't because the tensor product is not the Cartesian product.
The appropriate generalization is that $\text{FinVect}$ is a closed monoidal category. That means that it comes equipped with a natural isomorphism
$\text{Hom}(A \otimes B, C) \cong \text{Hom}(A, B \Rightarrow C)$
where we use $B \Rightarrow C$ to denote $\text{Hom}(B, C)$ treated as a vector space, to distinguish it from $\text{Hom}(B, C)$ treated as a set. (This distinction between internal hom and Hom may seem silly here, but in other categories it becomes important.) The above is in some sense just a restatement of the universal property of the tensor product, but in this form (at least for $R$-modules) it's usually called the tensor-hom adjunction, since it shows that $- \otimes B$ is left adjoint to $B \Rightarrow -$. (Incidentally this shows that the latter preserves colimits and the former preserves limits, a fact of great importance in homological algebra.)
The situation is slightly complicated when we talk about complex inner product places, so let's talk about real inner product spaces. Real inner product spaces $V$ come equipped with a canonical element of $\text{Hom}(V \otimes V, \mathbb{R})$, so plugging that canonical element into the above natural isomorphism gives a canonical element of $\text{Hom}(V, V^{\ast})$.
For a complex inner product space, the inner product is conjugate-linear in one variable, so you don't get a canonical morphism into the dual, you get a canonical anti-morphism into the dual (it's conjugate-linear).
What you're looking at seems like a natural transformation between functors, but I don't think it actually is because one of them (the identity functor) is covariant and the other (the dual space functor) is contravariant.