The traffic assignment problem is known as an optimization problem with a non-linear objective function and linear constraints, which allows one to estimate traffic congestion in an urban road network. Very early ideas on solving the traffic assignment problem concerned the concept of the existence of an equilibration operator. Such an operator was assumed to return assignment patterns that provide equal travel times on used routes between an origin-destination pair. Despite attempts to express this operator mathematically the first successful implementation of the idea came from computational techniques. The concept appeared to be fruitful and easy to apply from a computational perspective. Recently, a mathematical expression of the equilibration operator was found that generalized some of the previously obtained results. In the present paper, we discuss this operator and itemize key points to be paid attention to in the context of the equilibration operator application. Moreover, we demonstrate that the found operator, when applied to the problem, behaves very typically for path-equilibration techniques. The main advantage of the explicit form of the operator is the high accuracy of the solution. We believe this paper can clarify the matter and give fresh insights to traffic engineers. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.