It is proved that any category $\cal{K}$ which is equivalent to a simultaneously reflective and coreflective full subcategory of presheaves $[\cal{A}^{op},Set]$, is itself equivalent to the category of the form $[\cal{B}^{op},Set]$ and the inclusion is induced by a functor $\cal{A} \to \cal{B}$ which is surjective on objects. We obtain a characterization of such functors.
Moreover, the base category $Set$ can be replaced with any symmetric monoidal closed category $V$ which is complete and cocomplete, and then analogy of the above result holds if we replace categories by $V$-categories and functors by $V$-functors.
As a consequence we are able to derive well-known results on simultaneously reflective and coreflective categories of sets, Abelian groups, etc.
Keywords: monoidal category, reflection, coreflection, Morita equivalence.
2000 MSC: 18D20, 18A40.
Theory and Applications of Categories, Vol. 10, 2002, No. 16, pp 410-423.
http://www.tac.mta.ca/tac/volumes/10/16/10-16.dvi
http://www.tac.mta.ca/tac/volumes/10/16/10-16.ps
http://www.tac.mta.ca/tac/volumes/10/16/10-16.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/16/10-16.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/10/16/10-16.ps