My impression is that most, if not all, ''natural objects'' in linear algebra, analysis or differential geometry, ..., are usefully characterized by some \emph{symmetry} property, for eaxmple
''The exterior derivative is, up to a constant multiple, the only linear operator from $k$-forms to $k+1$-forms such that for each open embedding $f:U \to M$ and each form $\omega \in \Omega^k (M)$, the idenity $f^{\ast} d \omega = d (f^{\ast}\omega)$ holds.''
