  The High-Altitude Water Cherenkov Telescope (HAWC) has detected TeV halos
associated with two nearby pulsars/pulsar wind nebulae (PWN) -- Geminga and
B0656+14. These TeV halos extend up to tens of pc from the central
accelerators, indicating that the diffusion of ultrarelativistic electrons and
positrons in the interstellar medium has been suppressed by two orders of
magnitude. Although Geminga and B0656+14 are at similar distances and in the
same field of view, they have distinct histories. Notably, B0656+14 probably
still resides within its parent supernova remnant, the Monogem Ring, which can
be observed in X-rays. In this work, we perform high-resolution simulations of
the propagation and emission of relativistic lepton pairs around B0656+14 using
a two-zone diffusion model using the GALPROP numerical code. We compared the
predicted inverse-Compton spectrum to the observations made by HAWC and
Fermi-LAT and found physically plausible model parameters that resulted in a
good fit to the data. Additionally, we estimated the contribution of this
TeV-halo to the positron flux observed on Earth and found it to be smaller than
10\% of the measured flux. We conclude that future observations of the TeV halo
and its synchrotron emission counterpart in radio and X-ray frequencies will be
crucial to distinguish between various possible models.
