QGIS: keep symbology for an intersection

In QGIS, I am doing a shapefile intersection (Vector > Geoprocessing Tools > Intersect) to keep only the points in a point layer that overlap with the polygons in a polygon layer. What I’d like to do is have the option to save the original point layer symbology (which has been meticulously done and will be difficult to recreate) to the resulting intersected layer. However, the intersected layer always ends up with ‘default’ symbology. I’ve been searching around for the answer to this issue, but have not found it thus far.

Any pointer? I’m using QGIS 3.18 on a Windows 10 computer.