Identifying points under polygons after spatial joint

[MartíRosas]

Dear all,
I have posted this question in the general forum but without success. I am sure this is a pretty easy question and I am sorry if it has been answered before but I cannot find it anywhere. I am really desperate since I think it should be obvious and a usual geoprocessing need! The question is the following: I have a point layer (cities) and a polygon layer which are the polygon circular buffers of the aforementioned points for a specified distance (i.e., 10 km radii). I want to know which and how many of these cities fall within the limits of each buffer. If I perform a spatial joint, I will obtain (in the COUNT column) HOW MANY points I have under a certain polygon. But WHICH ARE THEY? I know I have, let's say, three points under polygon 4, but how can I obtain the OID's (or any other attribute for the case) of these points?
Thank you very much and cheers from Barcelona!

[LornaMurison]

You can use the "Intersect" tool.  A new point feature class will be created where each point will have the attributes of the original point and the attributes of the polygon that it intersects.  If one point falls within two polygons then that point will show up twice in the output, once with the attributes of each polygon.

Hope this helps.

[DarrenWiens2]

Just FYI, the Spatial Join tool (but not spatial join from the layer's right-click menu) will produce the same result as Intersect, using the points as the target layer, polygons as the join features, and the JOIN_ONE_TO_MANY option.

[HananAkilan]

I did the same and I got wrong number of points under joint count column. What might be the problem?