??so ... ??todd seems to be more or less suggesting to invent analog of "grothendieck topology" and/or of "lawvere-tierney topology" for tag doctrine ...

hmm... or i might be over-interpreting a bit here... maybe in general there's no really nice analog of these...

??what makes looking at what happens to the subterminal objects sufficient in certain cases to see what happens to the general objects?? ... hmmm...

??what about something about certain factorization systems here??? ...

there's various other possibly relevant doctrines here... for example just plain cocomplete categories ... ??also maybe some sort of categories with topos _and_ tag structure?? .... ??...

??something about reflective subcategories and gabriel-ulmer duality and so forth ... ??...

??cocontinuous monad on a presheaf category = ... ???

??algebra for it = ... ???

??and so forth ... ???....