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comma category
Naive notion of a comma category
Let $\mathcal{C}$ be a category and two subcategories $D_{1}$ and $D_{2}$ of $\mathcal{C}$. We form a category $(D_{1},D_{2})$ as follows:
1. 2. The morphisms of $(D_{1},D_{2})$ are of the form $(x,y):(a,b,f)\to(c,d,g)$, where $x:a\to c$ and $y:b\to d$ are morphisms, such that
$\xymatrix{a\ar[d]_{f}\ar[r]^{x}&c\ar[d]^{g}\\ b\ar[r]^{y}&d}$ is a commutative diagram: $yf=gx$.
It is easy to check that $(D_{1},D_{2})$ is indeed a category. For example, given object $(a,b,f)$, $(1_{a},1_{b})$ is the corresponding identity morphism. Furthermore, if we have the following two commutative diagrams
$\xymatrix{a\ar[d]_{f}\ar[r]^{x}&c\ar[d]^{g}\\ b\ar[r]^{y}&d}\qquad\qquad\qquad\qquad\xymatrix{c\ar[d]_{g}\ar[r]^{i}&m\ar[d]^% {h}\\ d\ar[r]^{j}&n}$ 
we may combine them and form the following commutative diagram
$\xymatrix{a\ar[d]_{f}\ar[r]^{x}&c\ar[d]_{g}\ar[r]^{i}&m\ar[d]^{h}\\ b\ar[r]^{y}&d\ar[r]^{j}&n}$ 
which shows that $h(ix)=(jy)f$, so that $(ix,jy)\in(D_{1},D_{2})$ is the composition of $(x,y)$ and $(i,j)$.
Definition. $(D_{1},D_{2})$, so constructed, is called a comma category (the comma between $D_{1}$ and $D_{2}$), or a slice category.
Examples. In the following examples, identity morphisms and compositions of morphisms are implicitly assumed to be included in the subcategories.

$D_{1}$ and $D_{2}$ each consists of a single object of $\mathcal{C}$, say $a,b$, then $(D_{1},D_{2})$ is just $\hom(a,b)$. The objects of $\hom(a,b)$ are just morphisms $a\to b$, and the morphism of $\hom(a,b)$ is $(1_{a},1_{b})$. $\hom(a,b)$ is a discrete category.

Similarly, we can form an “inverted cone” $(\mathcal{C},b)$ or $(\mathcal{C}\downarrow b)$.

Taking $D_{1}=D_{2}=\mathcal{C}$, then the comma category $(\mathcal{C},\mathcal{C})$ is the arrow category of $\mathcal{C}$ whose objects are morphisms of $\mathcal{C}$ and morphisms can be identified with commutative squares in $\mathcal{C}$.
Formal definition of a comma category
The diagrams $D_{1},D_{2}$, can be joined to the original category $\mathcal{C}$ via the inclusion functors $I_{1},I_{2}$:
$\xymatrix{D_{1}\ar[r]^{{I_{1}}}&\mathcal{C}&D_{2}\ar[l]_{{I_{2}}}}$ 
which suggests that a comma category may be more generally defined in terms of a pair of categories $\mathcal{A},\mathcal{B}$, and a pair of functors $F,G$ into a certain given category $\mathcal{C}$. Specifically, let $\mathcal{A}$ and $\mathcal{B}$ be categories and $F:\mathcal{A}\to\mathcal{C}$ and $G:\mathcal{B}\to\mathcal{C}$ be functors into a specific category $\mathcal{C}$. A comma category of the diagram
$\xymatrix{\mathcal{A}\ar[r]^{{F}}&\mathcal{C}&\mathcal{B}\ar[l]_{{G}}}$ 
written $(F,G)$ or $(F\downarrow G)$, consists of the following:
1. objects have the form $(a,b,f)$, where
(a) $a$ is an object of $\mathcal{A}$,
(b) $b$ is an object of $\mathcal{B}$, and
(c) $f:F(a)\to G(b)$ is a morphism in $\mathcal{C}$;
$\xymatrix{F(a)\ar[d]_{f}\\ G(b)}$
2. morphisms from $(a,b,f)$ to $(c,d,g)$ have the form $(x,y)$, where
(a) $x:a\to c$ is a morphism of $\mathcal{A}$,
(b) $y:b\to d$ is a morphism of $\mathcal{B}$, such that
(c) the following diagram
$\xymatrix@+=2cm{F(a)\ar[d]_{f}\ar[r]^{{F(x)}}&F(c)\ar[d]^{g}\\ G(b)\ar[r]^{{G(y)}}&G(d)}$ is commutative: $F(y)f=gF(x)$.
3. morphism composition in $(F\downarrow G)$ is given by $(x_{2},y_{2})\circ(x_{1},y_{1}):=(x_{2}\circ x_{1},y_{2}\circ y_{1})$, where $(x_{1},y_{1}):(a_{1},b_{1},f_{1})\to(a_{2},b_{1},f_{2})$ and $(x_{2},y_{2}):(a_{2},b_{2},f_{2})\to(a_{3},b_{3},f_{3})$.
It is an easy exercise to verify that indeed, $(F\downarrow G)$ is a category. For example, the identity morphism on $(a,b,f)$ is provided by the morphism $(1_{a},1_{b})$.
Remark. If $\mathcal{A}$ and $\mathcal{B}$ happen to be subcategories of $\mathcal{C}$ and $F,G$ are the inclusion functors, then we may write $(F,G)$ as $(\mathcal{A},\mathcal{B})$ or $(\mathcal{A}\downarrow\mathcal{B})$.
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