  Beyond-$\Lambda$CDM models have been proposed to address various shortcomings
of the standard cosmological model, such as the ``Hubble tension.'' These
models often have an impact on the discrepancy in the amplitude of matter
clustering, the ``$\sigma_8$-tension.'' To explore the interplay between the
two tensions, we suggest a simple method to visualize the relation between the
two parameters: $H_0$ and $\sigma_8$. For a given extension of the $\Lambda$CDM
model and dataset, we plot the relation between $H_0$ and $\sigma_8$ for
different amplitudes of the beyond-$\Lambda$CDM physics. In this work, we use
this visualization method to illustrate the trend of selected cosmological
models, including nonminimal Higgs-like inflation, early dark energy, a varying
effective electron mass, an extra number of relativistic species and modified
dark energy models. Although already studied in the literature, some of these
models have not been analyzed in view of the two joint tensions. We stress that
the method used here could be a useful diagnostic tool to illustrate the
behavior of complex cosmological models with many parameters in the context of
the $H_0$ and $\sigma_8$ tensions.
