hence as a quotient set of the the set of morphism in from into one of the objects . Being a quotient set, every element of it is represented by one of the original elements in .
This means that we have
The object commutes with the colimit over precisely if every morphism lifts to a morphism into one of the , schematically:
Let be a regular cardinal greater than . Then any -filtered category is also -filtered. For being -filtered means that any diagram in of size has a cocone; but any diagram of size is of course also . Thus, any -filtered colimit is also a -filtered colimit, so any functor which preserves -filtered colimits must in particular preserve -filtered colimits. It follows that any -presentable object is also -presentable.