Hrbacek and Jech gives the following definition of cardinal addition:
My question is: given an indexed system of cardinals $\left \langle \kappa_{i} |i\in I \right \rangle$ does there exist a system $\left \langle A_{i} |i\in I \right \rangle$ of mutually disjoint sets such that $|A_{i}|=\kappa_{i}$ for all $i \in I$?
Yes. Just let $A_i=\{i\}\times \kappa_i$, for instance.