Reading Awodey [p.16-17], he states the following:
The slice category $\boldsymbol{C}/C$ of a cateogry $\boldsymbol{C}$ over an object $C\in\boldsymbol{C}$ has [definition of slice category follows] (...)
If $g: C\to D$ is any arrow, then there is a composition functor, $g_{*}: \boldsymbol{C}/C \to \boldsymbol{C}/D$ defined by $g_{*}(f) = g\circ f$, and similarly for arrows in $\boldsymbol{C}/C$.
Indeed, the whole construction is a functor, $\boldsymbol{C}/(-): \boldsymbol{C} \to \boldsymbol{\operatorname{Cat}}$ as the reader can easily verify.
So I have a few questions:
- What does the $(-)$ symbol mean?
- What does the author mean by the expression "the whole construction"?
- And how is that "construction" a functor? To my best understanding slice category was a "category" and not a functor.
P.S.: Please let me know if it's not clear, and I'll expand/clarify.
P.S.S.: My mathematics level: newbie