In reply to Michel Hebert:
...it seems to me that the existence of a multi-initial object (in the sense of Diers) does not imply the existence of an initial object in each slice, unless you assume the existence of equalizers.
No, the other way round. The category of algebraically closed fields does have an initial object in each slice, alias initial candidates, alias a polyinitial family. The "poly" initial (etc) family is "multi" iff the relevant functor preserves equalisers. As I have said, all of these things become much simpler if you consider slices instead of these confusing collections (multiple-valued adjoints).
Doesn't this permit to simplify the definition of a locally-finitely poly-presentable category (as an accessible category with pullbacks)?
Indeed it does. I'm afraid my own work on this never got beyond the "notes" stage, but those notes are available by anonymous ftp from theory.doc.ic.ac.uk as /theory/papers/Taylor/LFPP.dvi Probably Pierre Ageron or Francois Lamarche has an account of the same thing. I don't like the term "locally finitely presentable", but given that after twenty years there's no changing it, "locally finitely multi- or poly- presentable" describes the corresponding categories with multi- or poly- (finite) colimits. There is a characterisation of the connections between how "poly" the colimits are in terms of finite limits in the same notes. Paul =====================================================================