Definition 17.1.4 (Category of Compatible Couples).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$ and $(E_{0}, E_{1})$ be a compatible couple, then $E_{0} E_{1}$ are a compatible couple in $\catc$ if $E_{0}, E_{1}, E_{0} \cap E_{1}, E_{0} + E_{1} \in \catc$.

Let $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$ be compatible couples in $\catc$ and $T \in L(E_{0} + E_{1}, F_{0} + F_{1})$, then $T$ is a morphism of compatible couples if $T|_{E_0}\in \text{Mor}_{\catc}(E_{0}; F_{0})$ and $T|_{E_1}\in \text{Mor}(E_{1}; F_{1})$.

The collection $\catc_{1}$ of all compatible couples in $\catc$ equipped with the above definition of morphisms is the category of compatible couples in $\catc$.