Summary: For supersymmetric fermion-boson models, the Nicolai map is so designed as (i) to cancel the fermion determinant and (ii) to reduce the boson measure to a free one. By analyzing a simple model, it is shown that (i) is realized only for suitable matrix elements of the evolution operator, but not for the trace; the realization requires proper formulations of the fermion path integral and the stochastic differential equation, which defines the map, with due attention paid to boundary conditions.