  We study a novel dark matter production mechanism based on the freeze-in
through semi-production, i.e. the inverse semi-annihilation processes. A
peculiar feature of this scenario is that the production rate is suppressed by
a small initial abundance of dark matter and consequently creating the observed
abundance requires much larger coupling values than for the usual freeze-in. We
provide a concrete example model exhibiting such production mechanism and study
it in detail, extending the standard formalism to include the evolution of dark
matter temperature alongside its number density and discuss the importance of
this improved treatment. Finally, we confront the relic density constraint with
the limits and prospects for the dark matter indirect detection searches. We
show that, even if it was never in full thermal equilibrium in the early
Universe, dark matter could, nevertheless, have strong enough present-day
annihilation cross section to lead to observable signals.
