What are the products in the category of normed vector spaces with linear contractions?

In the category of normed vector spaces in which the morphisms are linear contractions, what do products look like?

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