  We introduce and discuss the notion of naturally full functor. The definition
is similar to the definition of separable functor: a naturally full functor is
a functorial version of a full functor, while a separable functor is a
functorial version of a faithful functor. We study general properties of
naturally full functors. We also discuss when functors between module
categories and between categories of comodules over a coring are naturally
full.
