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 question #23188.