In the introduction to the reprint of "Metric spaces, generalized logic and closed categories" Lawvere talks about the following situation:

Let $\mathbb R_+$ denote $\mathbb R_{\geq 0}^\infty$. Every lax monoidal endofunctor $\lambda:\mathbb R_+\to\mathbb R_+$ gives rise to an endofunctor $\lambda.:\mathrm{Met}\to\mathrm{Met}$ on the category of generalized metric spaces, again leading to the notion of $\lambda$-Lipschitz-continous maps
$$ Lip^\lambda(X,Y):=Lip^1(X,\lambda.Y).$$
These are exactly the maps $f:X\to Y$ satisfying $d(x,x')\geq \lambda(d(fx,fx'))$. Lawvere goes on and "*suggest a whole family of monoidal structures [on the category of metric spaces] interpolating between*" the sum- (aka tensor product) and the max- (aka cartesian product) metric on the product of the underlying sets. He finally relates this to the
$$\frac{1}{p}+\frac{1}{q}=1$$
business occuring in analysis.

**Now for the question**: Has this been worked out somewhere (replacing $\mathbb R_+$ by an arbitrary moinoidal category $\mathcal V$)?

**Remark**: Concerning the monoidal structures my first idea was to define various adjoints to the hom-like functors
$$Lip^\lambda(X,-)$$
but as we don't have
$$Lip^\lambda\times Lip^\lambda\to Lip^\lambda$$
but rather
$$Lip^\lambda\times Lip^\mu\to Lip^{\mu\circ\lambda}$$
i suspect we should define $X\otimes_\lambda^\mu Y$ by an expression like
$$Lip^\lambda(X,Lip^\mu(Y,Z))=:Lip^\lambda(X\otimes_\lambda^\mu Y,Z)$$
(of maybe $Lip^1$ on the right hand side). So instead of various monoidal structures we'd get various tensor products, compatible in some way...