Isometric composition operators on Lipschitz spaces

Given pointed metric spaces $X$ and $Y$, we characterize the basepoint-preserving Lipschitz maps $\varphi$ from $Y$ to $X$ inducing an isometric composition operator $C_{\varphi }$ between the Lipschitz spaces $Li{p}_0(X)$ and $Li{p}_0(Y)$, 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].