No, not necessarily - the prompt-cache is only hit if everything prior matches... including MCP & tool definitions. So to benefit from prompt-caching themselves they would need to include the tools you define to the summariser.
If they were doing that, then a cheaper (prompt-cache friendly) option is just to silently send the whole conversation with the summariser prompt appended
If they were doing that, then a cheaper (prompt-cache friendly) option is just to silently send the whole conversation with the summariser prompt appended