TITLE: Using ends to construct categorical fixed points
QUESTION [13 upvotes]: Under advice from Toby Bartels, I am posting this question here; it falls under the general heading of constructing data types categorically as fixed points of functors.
The first question I have is a warm-up. There's a way to interpret a natural number n in any cartesian closed category $C$, as a dinatural transformation of the form
$$c^c \to c^c$$
which intuitively takes an element $f\colon c \to c$ to its $n$-th iterate $f^{(n)}\colon c \to c$. One may hope that if $C$ is "nice", then every such dinatural transformation will be of this form, or better still, that the end
$$\int_{c: C} (c^c)^{c^c}$$
(assuming it exists) behaves as a natural numbers object in $C$. So first, I am interested in what "nice" might mean here: what are some general conditions on $C$ that ensure we can construct a natural numbers object as an end in this way?
Second, this end can be rewritten as
$$\int_{c: C} c^{c^{1 + c}}$$
provided that $C$ has coproducts, and the assertion that this behaves as a natural numbers object is equivalent to saying it is an initial algebra for the endofunctor $F(c) = 1 + c$ (that's algebra for an endofunctor, not for a monad), making it a "fixed point" of $F$ by a famous old result of Lambek.
This suggests a second more general question: given an endofunctor $F: C \to C$ on cartesian closed $C$ with a strength (essentially, a structure of $C$-enrichment), I want to know what "nice" conditions on $C$ and/or $F$ guarantee that the end
$$e = \int_{c: C} c^{c^{F(c)}}$$
if it exists, is an initial $F$-algebra. It's not hard to write down an an $F$-algebra structure for this end $e$, and show that it is weakly initial, i.e., show that if $x$ is any $F$-algebra, then there at least exists an $F$-algebra map $e \to x$. The issue then is over the uniqueness of this map, or rather what nice conditions would guarantee that.
Discussion of specific cases like PER models would be alright, but I'd probably be a lot more excited if it led to consideration of more general abstract conditions on $C$ or $F$.

REPLY [2 votes]: I asked Marcelo Fiore about this last week, and he instantly recognized the type of problem --- right down to the fact that one typically has existence of a map from the supposedly-initial algebra to an arbitrary given algebra, but not necessarily uniqueness.  This answer consists of my best recollections of what Marcelo said.
The key phrase is "parametric polymorphism".  In the general setting of polymorphism, one has the existence property just mentioned, but not necessarily uniqueness.  In order to get uniqueness, one adds the extra ingredient of parametricity.  If I understand correctly (and I'm not confident that I do), adding in parametricity is something like passing from a category of objects and maps to the resulting category of objects and relations.  
One paper that Marcelo found was 
Categorical data types in parametric polymorphism by Ryu Hasegawa.  I think it wasn't quite what he was looking for, in that it doesn't contain any ends.  There are also relevant papers by both Freyd and Rosolini, apparently.
Sorry this is a bit garbled.