  We report the first observation of $a^{0}_{0}(980)$-$f_{0}(980)$ mixing in
the decays of $J/\psi\to\phi f_{0}(980)\to\phi
a^{0}_{0}(980)\to\phi\eta\pi^{0}$ and $\chi_{c1}\to a^{0}_{0}(980)\pi^{0}\to
f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$, using data samples of
$1.31\times10^{9}$ $J/\psi$ events and $4.48\times10^{8}$ $\psi(3686)$ events
accumulated with the BESIII detector. The signals of $f_{0}(980)\to
a^{0}_{0}(980)$ and $a^{0}_{0}(980)\to f_{0}(980)$ mixing are observed at
levels of statistical significance of $7.4\sigma$ and $5.5\sigma$,
respectively. The corresponding branching fractions and mixing intensities are
measured and the constraint regions on the coupling constants,
$g_{a_{0}K^{+}K^{-}}$ and $g_{f_{0}K^{+}K^{-}}$, are estimated. The results
improve the understanding of the nature of $a^{0}_{0}(980)$ and $f_{0}(980)$.
