Let $\Omega$ be the least uncountable ordinal. Let $\mathscr{K}(\Omega)$ be the category where the objects are the countable ordinals and where the morphisms are the strictly monotonic increasing functions. A dilator is a functor on $\mathscr{K}(\Omega)$ which preserves direct limits and pullbacks. Let $\tau < \Omega^E := \min\{\xi > \Omega : \xi = \omega^\xi\}$. Then $\tau$ has a unique "term"-representation in $\Omega. \lambda\xi\eta.\omega^\xi + \eta$ and countable ordinals called the constituents of $\tau$. Let $\delta < \Omega$ and $K(\tau)$ be the set of the constituents of $\tau$. Let $\beta = \max K(\tau)$. Let $\lbrack \beta \rbrack$ be an occurrence of $\beta$ in $\tau$ such that $\tau \lbrack \beta\rbrack = \tau$. Let $\bar \theta$ be the fixed point-free version of the binary Aczel-Buchholz-Feferman-function (which is defined explicitly in the text below) which generates the Bachman-hierarchy of ordinals. It is shown by elementary calculations that $\xi \mapsto \bar \theta(\tau \lbrack \gamma + \xi \rbrack)\delta$ is a dilator for every $\gamma > \max\{\beta. \delta.\omega\}$.