A functor $C\to Set$ is prorepresentable (or pro-representable) if it is a small filtered colimit of representables. In other words, it corresponds to a pro-object.

As these functors are ‘exactly’ the left exact functors, (at least with a caveat on size), that latter term can also be used, but there are some standard situations and conventions when this ‘pro-’ terminology is used.

Note

One tends to say ‘pro-object in $C$’, and may use various descriptions via the (system of) representing objects in $C$, but one does not usually say ‘prorepresentable functor in $C$’.

References

including

M. Artin, A. Grothendieck, J. L. Verdier. Théorie des topos et cohomologie étale des schemas. Lecture notes in mathematics 269, Springer-Verlag, Berlin.

A. Grothendieck, M. Raynaud et al. Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics 224, Springer 1971 (retyped as math.AG/0206203; published version Documents Mathématiques 3, Société Mathématique de France, Paris 2003)

M. Artin and B. Mazur, Étale homotopy theory, 1969, No. 100 in Lecture Notes in Maths., Springer-Verlag, Berlin.

For some applications see

Alexander I. Efimov, Valery Lunts?, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor, doi

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