  $ $Weak gravitational lensing is a powerful probe which is used to constrain
the standard cosmological model and its extensions. With the enhanced
statistical precision of current and upcoming surveys, high accuracy
predictions for weak lensing statistics are needed to limit the impact of
theoretical uncertainties on cosmological parameter constraints. For this
purpose, we present a comparison of the theoretical predictions for the
nonlinear matter and weak lensing power spectra, based on the widely used
fitting functions ($\texttt{mead}$ and $\texttt{rev-halofit}$), emulators
($\texttt{EuclidEmulator}$, $\texttt{EuclidEmulator2}$,
$\texttt{BaccoEmulator}$ and $\texttt{CosmicEmulator}$) and N-body simulations
($\texttt{Pkdgrav3}$). We consider the forecasted constraints on the $\Lambda
\texttt{CDM}$ and $\texttt{wCDM}$ models from weak lensing for stage III and
stage IV surveys. We study the relative bias on the constraints and their
dependence on the assumed prescriptions. Assuming a $\Lambda \texttt{CDM}$
cosmology, we find that the relative agreement on the $S_8$ parameter is
between $0.2-0.3\sigma$ for a stage III-like survey between the above
predictors. For a stage IV-like survey the agreement becomes $1.4-3.0\sigma$.
In the $\texttt{wCDM}$ scenario, we find broader $S_8$ constraints, and
agreements of $0.18-0.26\sigma$ and $0.7-1.7\sigma$ for stage III and stage IV
surveys, respectively. The accuracies of the above predictors therefore appear
adequate for stage III surveys, while the fitting functions would need
improvements for future stage IV weak lensing surveys. Furthermore, we find
that, of the fitting functions, $\texttt{mead}$ provides the best agreement
with the emulators. We discuss the implication of these findings for the
preparation of the future weak lensing surveys.
