  In a category with enough limits and colimits, one can form the universal
automorphism on an endomorphism in two dual senses. Sometimes these dual
constructions coincide, as in the categories of finite sets, finite-dimensional
vector spaces, and compact metric spaces. There, beginning with an endomorphism
$f$, there is a doubly-universal automorphism on $f$ whose underlying object is
the eventual image $\bigcap_n \mathrm{im}(f^n)$. Our main theorem unifies these
examples, stating that in any category with a factorization system satisfying
certain axioms, the eventual image has two dual universal properties. A further
theorem characterizes the eventual image as a terminal coalgebra. In all, nine
characterizations of the eventual image are given, valid at different levels of
generality.
