Proving that a slice category is still a category

Question

I recently came across the concept of a category, and I'm trying to make order of some stuff in my mind.

Given an object $c$ of a category $\mathbf C$, we define the slice category $c/\mathbf C$ to be the category where the objects are morphisms $c\to x$ of $\mathbf C$ and where arrows from $f\colon c\to x$ to $g\colon c\to y$ are morphisms $h\colon x\to y$ such that $g=hf$. Now, I have no idea how to show that the "thing" we built up this way is still a category. What does it mean to prove that the composition is unitary and associative in this case?

Answer 1

Remember that a category is comprised of data submitted to some axioms. The data are:

• a set/class of objects
• a set of morphisms for each given couple of objects
• a composition operation on morphisms
• a choice of identity morphisms

The axioms are the associativity law and the left and right unit laws.

Before proving any kind of law, you need the data! Here you have defined objects and morphisms, but not the composition and the identities. Once you have done that, you can move on to proving the associativity and unit laws (which you will see are basically trivial here).

Edit. As an analogous situation, imagine someone saying the following: given two groups $G$ and $G'$, I can for the set of pairs $(x,x')$ for $x\in G$ and $x'$ in $G'$, but I have no idea how to prove that this new object is a group! Well a group, say $H$, is supposed to have a operation $m:H\times H\to H$ and a constant $e\in H$ and none of that have been given for now for this "new object", so it surely can't be a group as such.