  What makes sets, or more precisely, the category {\bf Set} important in
Mathematics are the well known {\it two} specific ways in which arbitrary
mappings $f : X \longrightarrow Y$ between any two sets $X, Y$ can {\it fail}
to be bijections. Namely, they can fail to be injective, and/or to be
surjective. As for bijective mappings they are rather trivial, since with some
relabeling of their domains or ranges, they simply become permutations, or even
identity mappings. \\ To the above, one may add the {\it third} property of
sets, namely that, between any two nonvoid sets there exist mappings. \\ These
three properties turn out to be at the root of much of the interest which the
category {\bf Set} has in Mathematics. Specifically, these properties create a
certain {\it dynamics}, or for that matter, lack of it, on the level of the
category {\bf Set} and of some of its subcategories.
