  A dilator is a particularly uniform transformation $X\mapsto T_X$ of linear
orders that preserves well-foundedness. We say that $X$ is a Bachmann-Howard
fixed point of $T$ if there is an almost order preserving collapsing function
$\vartheta:T_X\rightarrow X$ (precise definition to follow). In the present
paper we show that $\Pi^1_1$-comprehension is equivalent to the assertion that
every dilator has a well-founded Bachmann-Howard fixed point. This proves a
conjecture of M. Rathjen and A. Montalb\'an.
