  We present a thorough analysis of the sequential freeze-in mechanism for dark
matter production in the early universe. In this mechanism the dark matter
relic density results from pair annihilation of mediator particles which are
themselves produced by thermal collisions of standard model particles. Below
some critical value of the mediator coupling to standard model fields, this
sequential channel dominates over the usual freeze-in where dark matter is
directly produced from thermal collisions, even when the mediator is not in
thermal equilibrium. The latter case requires computing the full non-thermal
distribution of the mediators, for which finite temperature corrections are
particularly important.
