  Given an upward directed set $I$ we consider surjective $I$-inverse systems
$\{X_\al,f_{\al\be}:X_\be\lra X_\al| \al\leq\be\in I\}$, namely those inverse
systems that have all $f_{\al\be}$ surjective. A number of properties of
$I$-inverse systems have been investigated; such are the Mittag-Leffler
condition, investigated by Grothendieck and flabby and semi-flabby $I$-inverse
systems studied by Jensen. We note that flabby implies semi-flabby implies
surjective implies Mittag-Leffler. Some of the results about surjective inverse
systems have been known for some time. The aim of this note is to give a series
of equivalent statements and implications involving surjective inverse systems
and the systems satisfying the Mittag-Leffler condition, together with
improvements of established results, as well as their relationships with the
already known, but scattered facts. The most prominent results relate
cardinalities of the index sets with right exactness of the inverse limit
functor and the non-vanishing of the inverse limit -- connections related to
cohomological dimensions.
