  We prove that adiabatic regularization and DeWitt-Schwinger point-splitting
provide the same result for the renormalized expectation values of the
stress-energy tensor for spin-$1/2$ fields. This generalizes the equivalence
found for scalar fields, which is here recovered in a different way. We also
argue that the coincidence limit of the DeWitt-Schwinger proper time expansion
of the two-point function exactly agrees with the analogous expansion defined
by the adiabatic regularization method at any order (for both scalar and
spin-$1/2$ fields). We also illustrate the power of the adiabatic method to
compute higher order DeWitt coefficients in FLRW universes.
