I've got a question about the following two categories: \begin{align*} x_0 \rightarrow y \leftarrow x_1, && 0 \rightarrow 1, \end{align*} the first called $ D $ and the second called $ 2 $. Then there is a unique functor $ F : D \Rightarrow 2 $ mapping objects as below, \begin{align*} F : D &\to 2 \\\\ x_0 &\mapsto 0 \\\\ x_1 &\mapsto 0 \\\\ y &\mapsto 1 . \end{align*} Now $ F $ seems to be
- full,
- faithful, and
- (essentially) surjective.
Hence $ F $ defines an equivalence of categories.
But $ D $ and $ 2 $ are equivalent iff $ \mathrm{sk}(D) \cong \mathrm{sk}(2) $, i.e. the two skeletons are isomorphic. But $ D = \mathrm{sk}(D) $ and $ 2 = \mathrm{sk}(2) $ and no such isomorphism exists... right? So what's going on? Any help appreciated!
$\hom(x_0, x_1) \to \hom(0, 0)$ isn't surjective, so $F$ isn't full.