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Highlights • In contradiction to the Debye relaxation the time composition of non-Debye relaxation processes can not be given by multiplication and to preserve the causality must be replaced by some more general rule. • We construct explicitly an example of such a rule for the Cole-Cole relaxation described in the time domain by the Mittag-Leffler function. • The time composition law for the Cole-Cole relaxation leads to the time evolution equations involving fractional derivatives. • It provides us with a meaningful argument that the natural language to used in investigations of processes with memory is that of fractional calculus. • The method used to represent the time evolution law for the Cole-Cole relaxation suggests how to analyze and to solve analogous problem for more general models of relaxation processes, e.g. the Havriliak-Negami model, and how to find suitable evolution equations. Abstract Physically natural assumption says that any relaxation process taking place in the time interval [ t 0 , t 2 ] , t 2 > t 0 ≥ 0 may be represented as a composition of processes taking place during time intervals [ t 0 , t 1 ] and [ t 1 , t 2 ] where t 1 is an arbitrary instant of time such that t 0 ≤ t 1 ≤ t 2. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain. [ABSTRACT FROM AUTHOR]