  A simple criterion for a functor to be finitary is presented: we call $F$
finitely bounded if for all objects $X$ every finitely generated subobject of
$FX$ factorizes through the $F$-image of a finitely generated subobject of $X$.
This is equivalent to $F$ being finitary for all functors between `reasonable'
locally finitely presentable categories, provided that $F$ preserves
monomorphisms. We also discuss the question when that last assumption can be
dropped. The answer is affirmative for functors between categories such as Set,
K-Vec (vector spaces), boolean algebras, and actions of any finite group either
on Set or on K-Vec for fields K of characteristic 0.
  All this generalizes to locally $\lambda$-presentable categories,
$\lambda$-accessible functors and $\lambda$-presentable algebras. As an
application we obtain an easy proof that the Hausdorff functor on the category
of complete metric spaces is $\aleph_1$-accessible.
