In SBE projections are isolated but there are cases where there are many projections sharing the same prefix e.g:

$project: {
     "d" : "a.b.c.d",
     "e" : "a.b.c.e",
     "f" : "a.b.c.f",
     "g" : "a.b.c.g",
     "h" : "a.b.c.h",
}

In the above example all the work to navigate up until c is repeated and thrown away on every projected field.

See the attached flame graph, query and explain plan for a real life example derived from HELP-47436