| Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. | |
| Let $F:\mathcal{C}\to\mathcal{D}$ be a functor. | |
| Then $F$ has a quasi-inverse if and only if | |
| \begin{enumerate} | |
| \item $F$ is fully faithful; | |
| \item $F$ is essentially surjective. | |
| \end{enumerate} |