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Current and upcoming cosmological experiments open a new era of precision cosmology, thus demanding accurate theoretical predictions for cosmological observables. Because of the complexity of the codes delivering such predictions, reaching a high level of numerical accuracy is challenging. Among the codes already fulfilling this task, $\textsf{PyCosmo}$ is a Python based framework providing solutions to the Einstein-Boltzmann equations and accurate predictions for cosmological observables. In this work, we first describe how the observables are implemented. Then, we check the accuracy of the theoretical predictions for background quantities, power spectra and Limber and beyond-Limber angular power spectra by comparison with other codes: the Core Cosmology Library ($\texttt{CCL}$), $\texttt{CLASS}$, $\texttt{HMCode}$ and $\texttt{iCosmo}$. In our analysis we quantify the agreement of $\textsf{PyCosmo}$ with the other codes, for a range of cosmological models, monitored through a series of $\textit{unit tests}$. $\textsf{PyCosmo}$, conceived as a multi purpose cosmology calculation tool in $\texttt{Python}$, is designed to be interactive and user friendly. A current version of the code (without the Boltzmann Solver) is publicly available and can be used interactively on the platform $\textsf{PyCosmo Hub}$, all accessible from this link: https://cosmology.ethz.ch/research/software-lab/PyCosmo.html . On the hub the users can perform their own computations using $\texttt{Jupyter Notebooks}$ without the need of installing any software, access to the results presented in this work and benefit from tutorial notebooks illustrating the usage of the code. The link above also redirects to the code release and documentation.