The slice category $\boldsymbol{C}/C$ of a cateogry $\boldsymbol{C}$ over an object $C\in\boldsymbol{C}$ has
- Objects: All arrows $f\in \boldsymbol{C}$ such that $cod(f) = C$,
- Arrows: an arrow $a$ from $f:X\to C$ to $f':X'\to C$ is an arrow $a:X\to X'$ in $\boldsymbol{C}$ such that $f'\circ a=f$
My question is how do we know that such an arrow as $a$ exists in $\boldsymbol{C}$? Or is this saying that if it exists, than we get arrows in slice category, and if it doesn't, our slice category can potentially be without any arrows?