"Representability" of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$? In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic functors.
I am confused about the problem with the example of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$. First of all, what is the "extra structure" on $S^1$ and why is it needed for representability of $\pi_1$? The comment by Tilman points out $\mathsf{hTop}_\ast$ doesn't have limits, but why does this matter? Whatever limits do exists should be preserved, no? Does he mean that these limits do not coincide with homotopy limits on $\mathsf{Top}_\ast$, which are the geometrical constructions we're interested in?

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*By default, when you talk about representable functors they have to be functors to $\text{Set}$. If you want them to take values in a more refined category that can sometimes happen if the representing object itself has additional structure. Here $S^1$ has the structure of a cogroup (in the pointed homotopy category), which corresponds to the group structure on $\pi_1$ by the Yoneda lemma.

*Very few limits exist, and in any case what we want to compute are homotopy limits. 
