  We propose a reformulation of the convergence theorem of monotone numerical
schemes introduced by Zhang and Zhuo for viscosity solutions of path-dependent
PDEs, which extends the seminal work of Barles and Souganidis on the viscosity
solution of PDE. We prove the convergence theorem under conditions similar to
those of the classical theorem in the work of Barles and Souganidis. These
conditions are satisfied, to the best of our knowledge, by all classical
monotone numerical schemes in the context of stochastic control theory. In
particular, the paper provides a unified approach to prove the convergence of
numerical schemes for non-Markovian stochastic control problems, second order
BSDEs, stochastic differential games etc.
