Parametric functions satisfy free theorems which state that they take related arguments to related results. This is formalized by the notion of parametric transformation introduced in section 5 of Logical relations and parametricty: A Reynolds programme for Category Theory and Programming Languages.
In a linear type system, I believe the evaluation function $ev_{AB} : (A \multimap B) \otimes A \to B$ should be parametric because it is defined independently of $A$ and $B$. Indeed, the paper states:
The “$evaluation$” map $ev_{AB} : [A \to B] \times A \to B$ given by $ev(f, x) = f (x)$ is similarly parametric in both $A$ and $B$ (as opposed to just $B$).
The reader will be able to construct similar examples for the internal homs in other closed categories.
However, I wasn't able to see how this result applies to general closed categories. In particular, it seems to fail for vector spaces. Fix some vector space $V$ and consider the evaluation map $ev_A : (A \multimap V) \otimes A \to V$.
Let $k$ be the base field and consider how $ev_k$ and $ev_0$ must be related. In particular, consider the linear relation $R : k \leftrightarrow 0$ which relates every $x \in k$ with $0$. Then I believe any $\varphi \otimes l \in (k \multimap V) \otimes k$ is related by $(R \multimap V) \otimes R$ to $0 \in (0 \multimap V) \otimes 0$. Then by parametricity, $ev$ sends related things to equal things in $V$. It must send $\varphi \otimes l$ to $0$, so it is not evaluation.
It seems like evaluation should still be parametric even when using linear types. Is there good explanation for why the expected free theorem fails to hold?