Definition 17.1.7 (Interpolation Functor).label Let $\catc$ be a subcategory of normed spaces over $K \in \RC$, $\catc_{1}$ be its categories of compatible couples, and $F: \catc_{1} \to \catc$ be a functor, then $F$ is an interpolation functor if for every $(E_{0}, E_{1}), (F_{0}, F_{1}) \in \catc_{1}$,

1. (1)

   $F((E_{0}, E_{1}))$ and $F((F_{0}, F_{1}))$ are interpolation spaces with respect to $(E_{0}, E_{1})$ and $(F_{0}, F_{1})$.

2. (2)

   For each $T \in \text{Mor}_{\catc_1}((E_{0}, E_{1}); (F_{0}, F_{1}))$, $F(T) = T|_{F((E_0, E_1))}$.