  We review the geometric theory of \emp{smooth systems of smooth maps}, of
\emp{smooth systems of smooth sections} of a smooth double fibred manifold and
of \emp{smooth systems of smooth connections} of a smooth fibred manifold.
  Moreover, after reviewing the concept of \emp{F-smooth space} due to A.
Fr\"olicher, we discuss the \emp{F-smooth systems of smooth maps}, the
\emp{F-smooth systems of fibrewisely smooth sections} of a smooth double fibred
manifold, the \emp{F-smooth systems of fibrewisely smooth connections} of a
smooth fibred manifold and of \emp{F-smooth connections} of an F-smooth system
of fibrewisely smooth sections.
