# Simultaneously Reflective And Coreflective Subcategories of Presheaves

## Robert El Bashir and Jiri Velebil

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.

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