I am intermittently thinking about categories again, and I think I got something wrong in the current set of notes. I there defined slice categories in a way that doesn’t work, at least given my initial preferred definition of categories (which is the same as Awodey’s or Riehl’s).
OK: What is an arrow in a slice category from the object to the object — where, of course, are -objects, and are -arrows?
Since we are constructing from data in the natural thing to do is to use a -arrow which interacts appropriately with and , giving us a commuting triangle with .
But that still leaves two options. The simpler option (1) identifies the needed -arrow with by itself. A more complex option (2) takes the needed -arrow to be the whole commuting triangle, or if you like, the triple . In the earlier set of notes I went for the simpler (1). But I don’t think this can be right for a reason I explain. And I now note that Leinster initially goes for (2) (though his language then wobbles).
So here are two improved(?)pages on slice categories which I do hope now get things right. But let me know if I have gone off-piste!