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PostPosted: Fri Nov 13, 2015 12:39 am 
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Let \( \mathcal{F} : \mathcal{C} \longrightarrow \mathcal{D} \) be a functor between the categories \( \mathcal{C} \) and \( \mathcal{D} \). Show that \( \mathcal{F} \) is an equivalence if and only if \( \mathcal{F} \) induces bijections on the morphism sets and, additionally, for every object \( \displaystyle D \) in \( \mathcal{D} \) there is an object \( \displaystyle C \) in \( \mathcal{C} \) such that \( \displaystyle \mathcal{F} \left( C \right) \cong D \).


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