I've been working with ESRI software for 28 years but I guess somewhere along the line I missed a memo or two Does the ESRI software no longer need the source and target feature classes in the same coordinate system for things like Append? A (younger) coworker told me that ArcGIS Desktop does the projection on the fly in behind the scenes automatically. If there are no reasons to have the source and target in the same projection we can eliminate the need for a staging copy when updating the Web Mercator copies we maintain for map services.
Does ArcGIS reproject on the fly in all of the Overlay tools too?
In the movement forward to obliterate human interaction in the thinking process and to foster the sense of 'automagically' I will allow you to check specific tools like ...Intersect... So as not to provide a direct answer but encourage all individuals to read the help topics, I will put a couple of snippets that might otherwise not be seen... The full details are in the help topic example
Determines the spatial reference for processing. This will also be the spatial reference of the output feature class. For details on how this is done, see Spatial Reference. All the input features are projected into this spatial reference for processing
cool you say.... some process...
feature type, but the spatial reference of the top entry on the tool dialog box (the first entry in scripting) in the list will be used during processing and set to the output.
but there is hope for the sentient being
To explicitly control the output spatial reference (coordinate system and domains), override the default Spatial Reference properties using the appropriate geoprocessing environments.
So in short, if you don't know what you are doing and you don't know what the coordinate systems are, then yes, you too can get output that looks like it worked but aren't sure what you have as an end produce. The important thing, is that you don't have to be involved in the thought process as long as there is an output...