Categories fibered in groupoids and the slice category

Question

I read that one example of categories fibered in groupoids is the slice category $\mathcal{C}_{/x}\to \mathcal{C}$ where $x\in \mathcal{C}$ an object. But as the usual definition of slice category goes I don't see why the fiber over any point has to be a groupoid. If there is a map $y\to z$ over $x$ it doesn't have to be an isomorphism in general. So can someone clarify, am I just being dumb as usual or is this probably a typo or something? (FYI : The source is http://arxiv.org/abs/1708.08124v1)

Answer 1

As Zhen Lin said in comment, this is actually fibered in sets.

The fiber of $\pi: \mathcal C_{/x} \to \mathcal C$ at $y$ is the category whose objects are arrows $y \to x$ and whose morphisms from $f: y \to x$ to $g: y \to x$ are morphisms $k$ from $f$ to $g$ in $\mathcal C_{/x}$ such that $\pi(k) = \mathrm{id}_y$. Now remember that a morphism $k$ from $f$ to $g$ in $\mathcal C_{/x}$ is simply an arrow of $\mathcal C$ such that $f=gk$.

I let you conclude that the only arrows in the fiber at $y$ are identities.