The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product:
Let $p \in [1,\infty]$ and let $(V_i)_{i \in I}$ be a family of Banach spaces over $\mathbb{K}=\mathbb{R},\mathbb{C}$. Then we may define a new Banach space $\bigoplus^p_{i \in I} V_i$ (how is this usually denoted?) as follows: The points are the sequences $(v_i)_{i \in I}$ with $v_i \in V_i$ and $\sum_{i \in I} \lVert v_i \rVert^p < \infty$. We let $\lVert (v_i)_{i \in I} \rVert := (\sum_{i \in I} \lVert v_i \rVert^p)^{1/p}$. For $p=\infty$ this has to be interpreted as $\lVert (v_i)_{i \in I} \rVert := \sup_{i \in I} \lVert v_i \rVert < \infty$. For $V_i=\mathbb{K}$ we get the usual space $l^p(I)$.
- If $p=1$, then $\bigoplus^1_{i \in I} V_i$ is the coproduct of $(V_i)_{i \in I}$.
- If $p=\infty$, then $\bigoplus^{\infty}_{i \in I} V_i$ is the product of $(V_i)_{i \in I}$.
Question. Does $\bigoplus^p_{i \in I} V_i$ have a useful universal property if $1<p<\infty$?
If necessary, you may work in a different category than $\mathsf{Ban_1}$. But would be nice if this category does not depend on $p$.