The comma category of two functors $f : C \to E$ and $g : D \to E$ is a category like an arrow category of $E$ where all arrows have their source in the image of $f$ and their target in the image of $g$ (and the morphisms between arrows keep track of how these sources and targets are in these images). It is a kind of 2-pullback: a directed refinement of the homotopy pullback of two functors between groupoids.
We discuss three equivalent definitions of comma categories
The terminology “comma category” is a holdover from the original notation $(f,g)$ for such a category, which generalises $(x,y)$ or $C(x,y)$ for a hom-set. This is rarely used any more. Another common notation for the comma category is $(f\downarrow g)$.
If $f:C\to E$ and $g:D\to E$ are functors, their comma category is the category $(f/g)$ whose
objects are triples $(c,d,\alpha)$ where $c\in C$, $d\in D$, and $\alpha:f(c)\to g(d)$ is a morphism in $E$, and whose
morphisms from $(c_1,d_1,\alpha_1)$ to $(c_2,d_2,\alpha_2)$ are pairs $(\beta,\gamma)$, where $\beta:c_1\to c_2$ and $\gamma:d_1\to d_2$ are morphisms in $C$ and $D$, respectively, such that $\alpha_2 . f(\beta) = g(\gamma) . \alpha_1$.
Let $I = \{a \to b\}$ be the (directed) interval category and $E^I = Funct(I,E)$ the functor category.
The comma category is the pullback
(in the 1-category Cat of categories).
Compare this with the construction of homotopy pullback (here), hence wth the definition of loop space object and also with generalized universal bundle.
Alternatively, the comma category is the “lax pullback” – or rather the comma object (see the discussion at 2-limit) of the pullback diagram,
i.e. the universal cone? that commutes up to a natural transformation
In terms of the imagery of loop spaces objects, the comma category is the category of directed paths in $E$ which start in the image of $f$ and end in the image of $g$.
If $f$ and $g$ are both the identity functor of a category $C$, then $(f/g)$ is the category $C ^{\mathbf{2}}$ of arrows in $C$.
If $f$ is the identity functor of $C$ and $g$ is the inclusion $1\to C$ of an object $c\in C$, then $(f/g)$ is the slice category $C/c$.
Likewise if $g$ is the identity and $f$ is the inclusion of $c$, then $(f/g)$ is the coslice category $c/C$.
The comma category $(f/g)$ comes with a canonical 2-cell in the square
which is universal in the 2-category Cat; that is, it is an example of a 2-limit (in fact, it is a strict 2-limit). Squares with the same universal property in an arbitrary 2-category are called comma squares and their top left vertex is called a comma object.
See at