Combining polygon features and retaining both geometries and attributes using `sf`

Question

I use the sf R package regularly, but I'm having trouble finding the correct process to use in sf to get a parallel result to what I would get performing a union in ArcGIS. Given the image below, where I have a set of polygon features in purple, and another polygon feature shown by the heavy black outline, what I want is what I would get from using union in ArcGIS - that is, the purple polygon features sliced on the border of the black polygon, and for the purple features within the black polygon, the attributes would now also contain attributes of my black polygon. For purple features outside the black polygon, they would have the attribute fields of the black polygon but they would have no values populated.

No matter how I parameterize st_union in sf, my interior purple polygons are dissolved with the bounding black polygon, and ALL features get all the attritues of the black polygon populated, even those outside the black polygon. What would be the analogous process in sf to get the result one would have using union in ArcGIS? It's simply a combination of both polygon features I need, where purple polygons are sliced by the black polygon, and features inside get attributes of black polygon feature, while those outside get the attribute fields of the black polygon but un-populated values.

Answer 1

Thanks @ElioDiaz, @Spacedman - here is code (could be simplified with chained operation) based on your suggestion that does exactly what I intended with trying to emulate result of union tool in ArcMap.

      outline <- read_sf('outline.shp') # black circle in picture
      other_polys <- read_sf('other_polys.shp') # purple polygons
      within <- st_within(other_polys, outline, sparse = FALSE)
      inside <- other_polys[within,] # interior to black polygon
      outside <- other_polys[!within,] # outside black polygon
      inside <- st_join(inside, outline) # attributes of black polygon added to interior polys
      missing_cols <- setdiff(names(inside), names(outside))
      outside[missing_cols] <- 0 # add columns of black polygon but with zeros for extiror purple polys
      full <- rbind(inside, outside) # join back together