  The exchange-only virial relation due to Levy and Perdew is revisited.
Invoking the adiabatic connection, we introduce the exchange energy in terms of
the right-derivative of the universal density functional w.r.t. the coupling
strength $\lambda$ at $\lambda=0$. This agrees with the Levy-Perdew definition
of the exchange energy as a high-density limit of the full exchange-correlation
energy. By relying on $v$-representability for a fixed density at varying
coupling strength, we prove an exchange-only virial relation without an
explicit local-exchange potential. Instead, the relation is in terms of a limit
($\lambda \searrow 0$) involving the exchange-correlation potential
$v_\mathrm{xc}^\lambda$, which exists by assumption of $v$-representability. On
the other hand, a local-exchange potential $v_\mathrm{x}$ is not warranted to
exist as such a limit.
