  Context. In the cooling process of a non-accreting neutron star, the
composition and properties of the crust are thought to be fixed at the finite
temperature where nuclear reactions fall out of equilibrium. A lower estimation
for this temperature is given by the crystallization temperature, which can be
as high as $\approx 7\times 10^9$ K in the inner crust, potentially leading to
sizeable differences with respect to the simplifying cold-catalyzed matter
hypothesis. Aims. We extend the recent work by Fantina et al. (2019) on the
outer crust, to the study of the crystallization of the inner crust and the
associated composition in the one-component plasma approximation. Methods. The
finite temperature variational equations for non-uniform matter in both the
liquid and the solid phases are solved using a compressible liquid-drop
approach with parameters optimized on four different microscopic models which
cover the present uncertainties in nuclear modeling. Results. We consider
separately the effect of the different nuclear ingredients with their
associated uncertainties, namely the nuclear equation of state, the surface
properties in the presence of a uniform gas of dripped neutrons, and the proton
shell effects arising from the ion single-particle structure. Our results
suggest that the highest source of model dependence comes from the smooth part
of the nuclear functional. Conclusions. We show that shell effects play an
important role at the lowest densities close to the outer crust, but the most
important physical ingredient to be settled for a quantitative prediction of
the inner crust properties is the surface tension at extreme isospin values.
