I am using ArcMap 10.4.1.
After watching the ESRI video presentation titled "Introducing Coordinate Systems and Transformations", under the subtitle "What is Happening When We Project Data?", under Case 2: Each PCS contains a different GCS; describes the projection as Un-projecting PCS A1 to GCS A->Transforming Geographic Datum to GCS B->then Projecting to PCS B1 as shown graphically in the attached screen shot.
After reviewing this information I am assuming that the ArcMap program is following this workflow, when the projection path is pre-defined. If the path is not pre-defined, then manually following this method is a logical solution. I have to ask if my understanding is correct; that the workflow presented may be a manual procedure if the projection path is not defined, following the above steps?