Let $M$ be a compact smooth manifold, let $\bar{g}$ be a fixed smooth Riemannian metric on $M$, and let $\hat{g}: [0,T) \to \Gamma(S^2T^*M)$ be a smooth family of Riemannian metrics solving Ricci-DeTurck flow with background $\bar{g}$. Let $\varphi: [0,T) \to \operatorname{Diff}(M)$ be the associated smooth family of DeTurck diffeomorphisms with $\varphi_0=\operatorname{id}_M$. Then the smooth family of metrics $g: [0,T) \to \Gamma(S^2T^*M)$ defined by

\begin{align*} g(t):=\varphi_t^*\hat{g}(t) \end{align*}

solves Ricci flow with $g(0)=\hat{g}(0)$.