The $\ell^p$-metrization of functors with finite supports

Let $p\in[1,\infty]$ and $F:\mathbf{Set}\to\mathbf{Set}$ be a functor with finite supports in the category $\mathbf{Set}$ of sets. Given a non-empty metric space $(X,d_X)$, we introduce the distance $d^p_{FX}$ on the functor-space $FX$ as the largest distance such that for every $n\in\mathbb N$ and $a\in Fn$ the map $X^n\to FX$, $f\mapsto Ff(a)$, is non-expanding with respect to the $\ell^p$-metric $d^p_{X^n}$ on $X^n$. We prove that the distance $d^p_{FX}$ is a pseudometric if and only if the functor $F$ preserves singletons; $d^p_{FX}$ is a metric if $F$ preserves singletons and one of the following conditions holds: (1) the metric space $(X,d_X)$ is Lipschitz disconnected, (2) $p=1$, (3) the functor $F$ has finite degree, (4) $F$ preserves supports. We prove that for any Lipschitz map $f:(X,d_X)\to (Y,d_Y)$ between metric spaces the map $Ff:(FX,d^p_{FX})\to (FY,d^p_{FY})$ is Lipschitz with Lipschitz constant $\mathrm{Lip}(Ff)\le \mathrm{Lip}(f)$. If the functor $F$ is finitary, has finite degree (and preserves supports), then $F$ preserves uniformly continuous function, coarse functions, coarse equivalences, asymptotically Lipschitz functions, quasi-isometries (and continuous functions). For many dimension functions we prove the formula $\dim F^pX\le\mathrm{deg}(F)\cdot\dim X$. Using injective envelopes, we introduce a modification $\check d^p_{FX}$ of the distance $d^p_{FX}$ and prove that the functor $\check F^p:\mathbf{Dist}\to\mathbf{Dist}$, $\check F^p:(X,d_X)\mapsto (FX,\check d^p_{FX})$, in the category $\mathbf{Dist}$ of distance spaces preserves Lipschitz maps and isometries between metric spaces.