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If you have an adjunction between two categories $\mathcal{C}$ and $\mathcal{D}$, and two full subcategories $\mathcal{\Gamma}$ and $\mathcal{\Delta}$ of $\mathcal{C}$ and $\mathcal{D}$ respectively, does that adjunction always restrict to an adjunction between $\mathcal{\Gamma}$ and $\mathcal{\Delta}$?

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Of course not. Try to think about any example ... –  Martin Brandenburg Aug 14 '13 at 9:05
    
However, the only problem is whether the functors restrict to said subcategories. –  Zhen Lin Aug 14 '13 at 9:55
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