Comma category

Remarks

The condition in clause 2 says that the square with sides $S(f)$, $h'$, $h$ and $T(g)$ is a commutative diagram; pasting two such squares shows that composites are again morphisms. Example: with $S = T = 1_{\mathcal{C}}$, the comma category is the arrow category of $\mathcal{C}$, whose objects are the morphisms $h \colon X \to Y$. In the arrow category of sets, the pair $(f, g)$ with $f = g = (x \mapsto -x)$ is a morphism from $h = (x \mapsto x^3)$ on $\R$ to itself, since $(-x)^3 = -x^3$. Non-example: for $h = (x \mapsto x^2)$ on $\R$, the same pair $(f, g)$ is not a morphism from $h$ to $h$, since $-x^2 \ne (-x)^2$ when $x \ne 0$. When $\mathcal{A}$ is the category with one object and one morphism, $S$ picks an object $X$ of $\mathcal{C}$ and one writes $(X \downarrow T)$. There are projection functors from $(S \downarrow T)$ to $\mathcal{A}$ and to $\mathcal{B}$.