Underlying the importance of communication in highly distributed and autonomous systems, i.e., multi-agent systems, such communication should be autonomously managed by the system itself. As such, it should be managed on the individual level of each individual agent, and still result in a general consistent framework of communication. Such an approach, opposite to the centralised and controlled stance, poses additional problems and introduces new challenges for the system design. It is therefore crucial to design and develop agents that could cope with this new tasks and be able to emerge, align and maintain a common framework of communication. This research intends to fill the current gap and investigate the dynamics of the model of individual semiosis, i.e., narrowing the interaction pattern of Language Game Model to a case of a single teaching agent. In particular, the presented research studies both, analytically and using a simulated framework, the dynamics of the alignment process itself, depending on the internal behaviour of the agent, and the dynamics of the observed phase transition in the alignment process in case of deviations from common context settings.