They can if given write access to "SOUL.md" (or "AGENT.md" or ".cursor" or whatever).

It's actually one of the "secret tricks" from last year, that seems to have been forgotten now that people can "afford"[0] running dozens of agents in parallel. Before everyone's focus shifted from single-agent performance to orchestration, one power move was to allow and encourage the agent to edit its own prompt/guidelines file during the agentic session, so over time and many sessions, the prompt will become tuned to both LLM's idiosyncrasies and your own expectations. This was in addition to having the agent maintain a TODO list and a "memory" file, both of which eventually became standard parts of agentic runtimes.

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[0] - Thanks to heavy subsidizing, at least.