In Homotopy Type Theory (HoTT in what follows) one may compute homotopy groups of objects that bear names that are the same as some usual spaces: for instance one may consider $S^1$ which is defined by $*:S^1, b : *=_{S^1}*$ and with the usual induction principle, and compute its homotopy groups.

It has been proved for instance that with this definition $\pi_1(S^1) = \mathbb{Z}$ and the higher homotopy groups vanish, which coincides with the usual homotopy groups of the usual $S^1$.

Now it seems that all the homotopy groups that have been computed in HoTT so far (say of spheres) coincide with the usual ones (in the sense that they have the same description : we can't literally compare them because they don't live in the same theory). My question is : how much of it is a coincidence ? I expect that it's *not* a coincidence, and so my real question is actually : can we explain it ? If so, in what terms ?

I'm not asking for a proof, but my problem is, the result seems even hard to state (as I said : the groups are not living in the same theory, and to begin how do we even relate HoTT's $S^1$ with the usual $S^1$ ? ).

So I would want an answer that explains how to *state* the result (not necessarily the details, but at least the main ideas), and *how* to prove it (again, the main ideas of proof - I expect something like "oh so we have this model of HoTT that we can define in set theory, and when you interpret HoTT in this model, $S^1$ looks like the circle and its homotopy groups look like the usual homotopy groups")