📝 clarify library async wrapper docs

docs/api/library-async-logger-to-async-logger.md

@@ -3,7 +3,7 @@ name: library-async-logger-to-async-logger
 group: api
 category: facade
 update-time: 20260613
-description: Recover the underlying full async logger from the library-facing async facade.
+description: Recover the underlying full async logger from the library-facing async facade without rebuilding queue or lifecycle state.
 key-word:
     - async
     - library
@@ -34,8 +34,9 @@ pub fn[S] LibraryAsyncLogger::to_async_logger(self : LibraryAsyncLogger[S]) -> A
 Detailed rules explaining key parameters and behaviors
 
 - This conversion unwraps the existing async logger instead of rebuilding it.
-- Queue state, sink wiring, target, and flush policy remain the same.
+- Queue state, sink wiring, target, min level, flush policy, and current failure/lifecycle state remain the same.
 - Use this when code needs wider async logger APIs outside the library facade.
+- This is the step required for helpers such as `pending_count()`, `dropped_count()`, `state()`, `wait_idle()`, `has_failed()`, `last_error()`, or broader composition methods that are intentionally hidden by `LibraryAsyncLogger[S]`.
 
 ### How to Use
 
@@ -45,12 +46,23 @@ Here are some specific examples provided.
 
 When a library-facing async logger must be widened for additional async logger composition:
 ```moonbit
-let library_logger = LibraryAsyncLogger::new(console_sink(), target="lib.async")
+let library_logger = LibraryAsyncLogger::new(@bitlogger.console_sink(), target="lib.async")
 let full_logger = library_logger.to_async_logger().with_timestamp()
 ```
 
 In this example, the facade is unwrapped so the caller can access the full async logger API again.
 
+#### When Need Direct Async State Inspection
+
+When library-facing code later needs runtime diagnostics from the wrapped logger:
+```moonbit
+let full = library_logger.to_async_logger()
+ignore(full.pending_count())
+ignore(full.state())
+```
+
+In this example, unwrapping exposes the broader async inspection helpers on the same underlying logger state.
+
 ### Error Case
 
 e.g.:
@@ -58,8 +70,12 @@ e.g.:
 
 - Unwrapping does not start or stop the background runtime by itself.
 
+- Recovering the full async logger does not clear queue contents or reset failure state.
+
 ### Notes
 
 1. Use this only when the narrower async facade is no longer sufficient.
 
 2. This is the inverse projection of `AsyncLogger::to_library_async_logger()`.
+
+3. Use `LibraryAsyncLogger[S]` directly when the narrower package-facing surface is still sufficient.