We study a model of fermionic dark matter interacting with the Standard Model through a sequential Z' mediator, the gauge boson of a U(1) extension to the Standard Model symmetry group, to understand the mechanism responsible for the dark matter relic abundance. We compare two different mechanisms for dark matter production in the early universe, freeze-out and freeze-in. For production through freeze-out, dark matter particles were in thermal equilibrium with the cosmic plasma until the expansion of the universe dominated over the interactions. For freeze-in, dark matter was never in thermal equilibrium with the visible sector, and its abundance is always negligible compared to the thermal bath abundances. The boundary between each production regime is explored by considering every free parameter of this model, namely the masses and couplings. Details of the boundary region between freeze-in and freeze-out could have implications for dark matter searches.