# Is this definition of $(c\downarrow G)$ the slice category or is it something else?

I'm learning about the slice and coslice category constructs, and I think I understand the basics from Wikipedia.

However, in this lecture script (in German), there's another definition given, which sounds similar, but I'm not sure if it's really the same, or just somehow related to the slice/coslice categories.

On page 20 (Beispiel 2.4.18), the comma category is defined:

Let $C,D,E$ be categories and $F:C\to E$ and $G:D\to E$ be functors. First, we define the graph $(F\downarrow G)$, which has as objects triples $(c,d,f)$ with objects $c\in C,d\in D$ and a morphism $f:F(c)\to G(d)$. Between two objects $(c,d,f),(c',d',f')$, the arrows are pairs of morphisms $(\alpha:c\to c',\beta:d\to d')$ such that the diagram [...] commutes. Through the component-wise composition of arrows we get the category $(F\downarrow G)$.

On the next page (Beispiel 2.4.19.(3)), the following special case is given:

If $C=E$ and $F:\{c\}\to C$ is the inclusion of one object $c\in C$, then we write $(c\downarrow G)$ instead of $(F\downarrow G)$.

This sounds very similar to the slice or coslice category $(\mathcal C\downarrow A)$ or $(A\downarrow\mathcal C)$ as defined in Wikipedia, but there are differences:

• In Wikipedia, one of the "source categories" is isomorphic to the target category while the other one is the terminal (one object, one morphism) category. In the lecture script, this "one-object inclusion functor" does not come from the terminal category but from the one that's isomorphic to the target category of both functors.
• In Wikipedia, the category under construction are noted as e.g. $(\mathcal C\downarrow A)$, which is a category and an object. In the lecture script, the notation is $(c\downarrow G)$, which is an object and a functor.

How are these definitions related to each other?

The distinction in your first bullet is not what actually occurs. As the notation says, the inclusion functor $F$ has domain the terminal category $\{c\}$. There's a typo in the lecture notes: your last quote should begin "If $D=E$."
The author also fails to make clear that in the special case $G$ is meant to be the identity functor of $D$ (and of $E=D$.) So the two definitions agree, that is, $(\mathcal C\downarrow A)=(\text{id}_\mathcal{C}\downarrow c)$, in Wiki's and the lecturer's notation respectively.