In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$ , for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

In this paper, we establish the following results: (i) every strongly finite dilator is finitely presentable in the category of endofunctors on the category of ordinals; (ii) a dilator F is strongly finite if and only if F is finitely presentable in the category of dilators.