  If C and D are varieties of algebras in the sense of general algebra, then by
a representable functor C --> D we understand a functor which, when composed
with the forgetful functor D --> Set, gives a representable functor in the
classical sense; Freyd showed that these functors are determined by D-coalgebra
objects of C. Let Rep(C,D) denote the category of all such functors, a full
subcategory of Cat(C,D), opposite to the category of D-coalgebras in C.
  It is proved that Rep(C,D) has small colimits, and in certain situations,
explicit constructions for the representing coalgebras are obtained.
  In particular, Rep(C,D) always has an initial object. This is shown to be
"trivial" unless C and D either both have_no_ zeroary operations, or both have
_more_than_one_ derived zeroary operation. In those two cases, the functors in
question may have surprisingly opulent structures.
  It is also shown that every set-valued representable functor on C admits a
universal morphism to a D-valued representable functor.
  Several examples are worked out in detail, and areas for further
investigation noted.
