  Given pointed metric spaces $X$ and $Y$, we characterize the
basepoint-preserving Lipschitz maps $\phi$ from $Y$ to $X$ inducing an
isometric composition operator $C_\phi$ between the Lipschitz spaces $Lip_0(X)$
and $Lip_0(Y)$, whenever $X$ enjoys the peak property. This gives an answer to
a question posed by N. Weaver in his book [Lipschitz algebras. Second edition.
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].
