PERF: pass how=left to _factorize_keys in _left_join_on_index (#65160)

doc/source/whatsnew/v3.1.0.rst

@@ -142,6 +142,7 @@ Performance improvements
 - Performance improvement in :func:`merge` for many-to-one joins with unique right keys (:issue:`38418`)
 - Performance improvement in :func:`merge` with ``how="cross"`` (:issue:`38082`)
 - Performance improvement in :func:`merge` with ``how="left"`` (:issue:`64370`)
+- Performance improvement in :func:`merge` with ``how="left"`` and ``sort=False`` when joining on a right index with unique keys (:issue:`65160`)
 - Performance improvement in :func:`merge` with ``sort=False`` for single-key ``how="left"``/``how="right"`` joins when the opposite join key is sorted, unique, and range-like (:issue:`64146`)
 - Performance improvement in :func:`read_csv` with ``engine="c"`` when reading from binary file-like objects (e.g. PyArrow S3 file handles) by avoiding unnecessary ``TextIOWrapper`` wrapping (:issue:`46823`)
 - Performance improvement in :func:`read_html` and the Python CSV parser when ``thousands`` is set, fixing catastrophic regex backtracking on cells with many comma-separated digit groups followed by non-numeric text (:issue:`52619`)

pandas/core/reshape/merge.py

@@ -2816,10 +2816,13 @@ def _left_join_on_index(
         # variable has type "ndarray[Any, dtype[signedinteger[Any]]]")
         rkey = right_ax._values  # type: ignore[assignment]
 
-    left_key, right_key, count = _factorize_keys(lkey, rkey, sort=sort)
-    left_indexer, right_indexer = libjoin.left_outer_join(
-        left_key, right_key, count, sort=sort
-    )
+    left_key, right_key, count = _factorize_keys(lkey, rkey, sort=sort, how="left")
+    if count == -1:
+        left_indexer, right_indexer = left_key, right_key
+    else:
+        left_indexer, right_indexer = libjoin.left_outer_join(
+            left_key, right_key, count, sort=sort
+        )
 
     if sort or len(left_ax) != len(left_indexer):
         # if asked to sort or there are 1-to-many matches