ct.category theory - What are the products in the category of normed vector spaces with linear contractions?

Two words: sup norm.

I.e., the product of a family is the uniformly bounded subset of the cartesian product of the family, and the norm is the smallest uniform bound.

Explicitly, if ${X_i}_{iin I}$ is a family of normed vector spaces with all norms ambiguously denoted $|cdot|$, then the product is $X={{a_i}inprod X_i:sup|a_i|<infty}$, and for ${a_i}in X$, $|{a_i}|=sup|a_i|$.

(My intuition came from products of C*-algebras, where the $*$-homomorphisms are automatically contractive and products are defined in this way. So I had a good guess and it is easy to check that it works.)