Consider the following fragment from the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset (p72, in section 2.5):

$\ \ \ $ Assume $\mathscr{C}$ is a category having all the properties of a strict $\scr C^*$-tensor category except existence of direct sums and subobjects. First complete it with respect to direct sums. For this, consider the category $\mathscr{C}'$ consisting of $n$-tuples $(U_1,\ldots,U_n)$ of objects in $\mathscr{C}$ for all $n\geqslant 1$. Morphisms are defined by $$\operatorname{Mor}\big((U_1,\ldots,U_n),(V_1,\ldots,V_m)\big)=\oplus_{i,j}\operatorname{Mor}(U_i,V_j).$$ Composition and involution are defined in the obvious way. Note that the norm is uniquely determined by the $\rm C^*$-condition $\|T\|^2=\|T^*T\|$, and for its existence we do need condition (ii) (c) in Definition 2.1.1 to be satisfied in $\mathscr{C}$. The tensor product of $(U_1,\ldots,U_n)$ and $(V_1,\ldots,V_m)$ is defined as the $nm$-tuple consisting of objects $U_i\otimes V_j$ ordered lexicographically. The category $\mathscr{C}'$ has direct sums: they can be defined by concatenation.

As I understand it, the morphism spaces between objects $(U_1, \dots, U_n)$ and $(V_1, \dots, V_m)$ just consist of matrices of morphisms between these objects, and composition and involution are defined as the natural matrix operations.

Tensoring morphisms corresponds to the formal Kronecker product of matrices.

What I do not understand however, is how these morphism spaces become Banach spaces (which is a requirement for these kinds of categories). There is a sentence in the text that mentions something about this: "Note that the norm .... to be satisfied in $\mathscr{C}$". However, I do not understand what the author is saying here.

Can someone clarify?