  Fix a metric space $M$ and let $\mathrm{Lip}_0(M)$ be the Banach space of
complex-valued Lipschitz functions defined on $M$. A weighted composition
operator on $\mathrm{Lip}_0(M)$ is an operator of the kind $wC_f : g \mapsto w
\cdot g \circ f$, where $w : M \to \mathbb C$ and $f: M \to M$ are any map.
When such an operator is bounded, it is actually the adjoint operator of a
so-called weighted Lipschitz operator $w\widehat{f}$ acting on the
Lipschitz-free space $\mathcal F(M)$. In this note, we study the spectrum of
such operators, with a special emphasize when they are compact. Notably, we
obtain a precise description in the non-weighted $w \equiv 1$ case: the
spectrum is finite and made of roots of unity.
