  By means of several examples, we motivate that universal properties are the
simplest way to solve a given mathematical problem, explaining in this way why
they appear everywhere in mathematics. In particular, we present the
co-universal property of Schwartz distributions, as the simplest way to have
derivatives of continuous functions, Colombeau algebra as the simplest quotient
algebra where representatives of zero are infinitesimal, and generalized smooth
functions as the universal way to associate set-theoretical maps of
non-Archimedean numbers defined by nets of smooth functions (e.g.
regularizations of distributions) and having arbitrary derivatives. Each one of
these properties yields a characterization up to isomorphisms of the
corresponding space. The paper requires only the notions of category, functor,
natural transformation and Schwartz's distributions, and introduces the notion
of universal solution using a simple and non-abstract language.
