This might be a dumb question.  If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits.  However, it can happen that $c$ can be equipped with extra structure which in turn gives the morphisms out of $c$ extra structure, so that there is a "representable functor" $\text{Hom}(c, -) : C \to D$ where $D$ is a category equipped with a forgetful functor $F : D \to \text{Set}$ such that composing with the above gives the original representable functor.  

In this situation, when does the functor into $D$ still preserve limits?  How is this situation formalized?  (Assume that $C$ is not enriched over $D$ in any obvious way.)  

There are several examples of this coming from algebra, but the one that got me curious is the following.  Let $C$ denote the homotopy category of pointed (path-connected?) topological spaces and let $S^1$ denote the circle with a distinguished point.  I believe I am correct in saying that if the fundamental group functor $\pi_1 : C \to \text{Grp}$ is composed with the forgetful functor $U : \text{Grp} \to \text{Set}$, then $S^1$ represents the resulting functor $U(\pi_1(-))$.  (The extra structure on $S^1$ that makes this possible is, if I'm not mistaken, a **cogroup** structure internal to $C$.)  Can I conclude that $\pi_1$ preserves limits?  

---

**Edit:**  I've been told that the above example is problematic, so here's a simpler one.  Let $C = \text{Set}$ and suppose that $c \in C$ is equipped with a morphism $f : c \to c$.  Then by precomposition $\text{Hom}(c, d)$ is also equipped with such a morphism, so $\text{Hom}(c, -)$ has values in the category of dynamical systems.  Does it preserve limits?  Another example is my attempted answer to <a href="https://mathoverflow.net/questions/23188/distributivity-of-unary-operations/23191#23191">question #23188</a>.