What I can say is that many interesting and nontrivial categories do arise in certain parts of functional analysis and it is useful to understand the structure of these categories. The specific part of functional analysis that I have in mind is the theory of operator algebras. For instance, in C*-algebra theory one considers a category whose objects are C*-algebras and whose morphisms are given by groups $KK(A,B)$ which simultaneously generalize K-theory and K-homology. Many of the deepest theorems in the subject are organized around the "Kasparov product" which is nothing more than the composition law $KK(A,B) \times KK(B,C) \to KK(A,C)$ in this category. KK-theory and its close cousin E-theory can be characterized according to homotopy invariance and various functorial properties.