Geodesic
Simultaneously Reflective And Coreflective Subcategories of Presheaves
Theory and applications of categories, Tome 10 (2002), pp. 410-423.

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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.
Classification : 18D20, 18A40.
Keywords: monoidal category, reflection, coreflection, Morita equivalence. 
@article{TAC_2002_10_a15,
     author = {Robert El Bashir and Jiri Velebil},
     title = {Simultaneously {Reflective} {And} {Coreflective} {Subcategories} of {Presheaves}},
     journal = {Theory and applications of categories},
     pages = {410--423},
     publisher = {mathdoc},
     volume = {10},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2002_10_a15/}
}
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Robert El Bashir; Jiri Velebil. Simultaneously Reflective And Coreflective Subcategories of Presheaves. Theory and applications of categories, Tome 10 (2002), pp. 410-423. http://geodesic.mathdoc.fr/item/TAC_2002_10_a15/