My question has to do with the following assetion given in J.M. Lee: Introduction to Topological Manifolds, page 364, in the context of demonstrating there exists a chain homotopy $h$ between the subdivision operator $s$ and $\mathcal{i}_p$, the identity in $\Delta_p$, defined by
$$h\sigma=\sigma_\#b_p*(\mathcal{i_p}-s\mathcal{i}_p-h\partial s\mathcal{i_p})$$
Observe also that if $\sigma$ is a $\mathcal{U}$-small simplex, then $h\sigma$ is a $\mathcal{U}$-small chain, so $h$ maps $C_p^\mathcal{U}(X)$ to $C_{p+1}^\mathcal{U}(X)$
How can we actually be sure that is the case? I searched for an answer in Glen E. Bredon: Topology and Geometry, but I found nothing.
It would be interesting if anyone could give more insight to what $h$ actually does $-$geometrically speaking$-$, since the definition by recurrence doesn't offer any intuition at all
Thanks in advance for your help.