  The recent detection of the Geminga PWN by HAWC in the multi-TeV band allows
us to infer precious information about the transport of pairs in the immediate
surroundings of the pulsar and on the spectrum of pairs contributed by a
Geminga-like pulsar to the spectrum of pairs in the cosmic radiation. Moreover,
this detection allows us to address the issue of how typical are the so-called
TeV halos associated to PWNe. Our calculations confirm the need to have
suppressed diffusion in a region of at least $20-50\,$pc around the pulsar, and
are used here to infer precious constraints on the spectrum of pairs
accelerated at the termination shock: more specifically, we discuss the
conditions under which such a spectrum is consistent with that typically
expected in a PWN and how it gets modified once it escapes the halo. Finally,
we discuss the implications of the existence of a TeV halo around Geminga in
terms of acceleration of protons in the pulsar environment, a topic of profound
relevance for the whole field of particle acceleration and physics of pulsars.
