Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds

In the lecture Notions of Scalar Curvature – IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":

Suppose $(X,g_X)$ and $(Y,g_Y)$ are Riemannian manifolds, their sectional curvature satisfy $sec(Y,g_Y)leq kappaleq sec(X,g_X)$ for some $kappainmathbb{R}$, and $X_0$ is a subset of $X$. If $f_0:X_0to Y$ is a map with Lipschitz constant $1$, then there exists a map $f:Xto Y$ with Lipschitz constant $1$ that extends $f_0$, i.e. $f|_{X_0}=f_0$.

He mentions a few names before stating the result, but I cannot make out who they are.

He then discusses how this can be used to motivate a definition of "curvature" in the category of metric spaces with distance non-increasing maps, "except, of course, for normalization."

Does anyone know where I can read more about this? (Either in the setting of metric spaces or in the smooth setting of Riemannian manifolds.)