  By investigating the vacancy and substitutional defects in monolayer WS$_2$
with hybrid functionals, we find that there is no unique amount of Fock
exchange that concurrently satisfies the generalized Koopmans' condition and
reproduces the band gap and band-edge positions. Fixing the mixing parameter of
Fock exchange based upon the band gap can lead to qualitatively incorrect
defect physics in two-dimensional materials. Instead, excellent agreement is
achieved with experiment and many-body perturbation theory within $GW$
approximation once the mixing parameters are tuned individually for the defects
and the band edges. We show the departure from a unique optimized mixing
parameter is inherent to two-dimensional systems as the band edges experience a
reduced screening whilst the localized defects are subject to bulklike
screening.
