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Example Let $[0]$ denote the category having a single object and a single morphism. For any category $\operatorname{\mathcal{E}}$, there is a unique functor $U: \operatorname{\mathcal{E}}\rightarrow [0]$. The following conditions are equivalent:

  • The functor $U$ is a left covering functor.

  • The functor $U$ is a right covering functor.

  • The category $\operatorname{\mathcal{E}}$ is discrete: that is, every morphism in $\operatorname{\mathcal{E}}$ is an identity morphism.