  Fast-growing hierarchies are sequences of functions obtained through various
processes similar to the ones that yield multiplication from addition,
exponentiation from multiplication, etc. We observe that fast-growing
hierarchies can be naturally extended to functors on the categories of natural
numbers and of linear orders. We show that the categorical extensions of binary
fast-growing hierarchies to ordinals are isomorphic to denotation systems given
by ordinal collapsing functions, thus establishing a connection between two
fundamental concepts in Proof Theory.
  Using this fact, we obtain a restatement of the subsystem $\Pi^1_1$-CA$_0$ of
analysis as a higher-type wellordering principle.
