Use ArcGIS to classify intersection types

I am using ArcMap 10.4

I have a shapefile of a road network. Each road has a functional class value in the ‘FC’ field. I’m trying to determine how often roads of each functional class intersects another.

1. I Exported a shapefile of each functional class (Definition queries interact oddly with intersect).
2. As some segments consisted of multiple bits, I Dissolved each, and then ran Multipart to Singlepart to get a file where each segment was one feature.
3. Some of the roads are divided highways, so I’ve used ‘Merge Divided Roads‘ and ‘Collapse Dual Lines to Centerline‘ at various distances. Neither has been completely satisfactory, but oh well.
4. Then I’ve run Intersect between the two shapefiles, setting the output to point.

But I’m uncertain how to get the data on which lines it intersects back to the point, so as to classify the point by the types of functional class of the roads intersection. Past process I’ve used has been to use Select, and field calculate four columns, but seems like their ought be a better way.