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## Category Theory for beginners (5)

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Tsakanikas Nickos
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### Category Theory for beginners (5)

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 $\mathcal{F}$ 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 $\displaystyle \mathcal{F} \left( C \right) \cong D$.