A lifting theorem for operators on spaces of Lipschitz functions

We prove that every bounded linear operator between Lipschitz spaces admits a lifting along the De Leeuw embedding. More precisely, given pointed metric spaces $M$ and $N$ and $\epsilon>0$, every bounded linear operator $S:\mathrm{Lip}_0(M)\to \mathrm{Lip}_0(N)$ admits a lifting $\mathfrak{S}:C(\beta \tilde{M})\to C(\beta \tilde{N})$ such that $\|\mathfrak{S}\|\leq \|S\|+\epsilon$ and $\mathfrak{S}(\varPhi_M(f))=\varPhi_N(S(f))$ for every $f\in \mathrm{Lip}_0(M)$.