[[Category theory MOC]]
# Slice category

Let $\cat C$ be a [[category]] and $C \in \Ob \cat C$.
The **slice category** $\cat C / C$ is defined as follows:[^2010] #m/def/cat 

- $f \in\ob (\cat C / C)$ for $X \in \Ob \cat C$ and $f \in \cat C(X,C)$.
- $a \in \cat C / C ((X,f), (X',f'))$ is a morphism $a \in \cat C(X,X')$ such that the following diagram commutes:

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[^2010]: 2010\. [[Sources/@awodeyCategoryTheory2010|Category theory]], p. 16

Typically the objects are referred to by the morphism (e.g. $f$) only. 
A slice category is a special case of a [[Comma category]].

## Properties
- There exists a functor $U : \cat C / C \to \cat C$ that forgets the base object.

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