nLab
comma object

The comma object of two morphisms $f:A\to C$ and $g:B\to C$ in a 2-category is an object $(f/g)$ equipped with projections $p:(f/g)\to A$ and $q:(f/g)\to B$ and a 2-cell

which is universal in the sense of a 2-limit.

Part of this (to be explicit) is the statement that for any object $D$, 1-morphisms $p\prime :D\to A$, $q\prime :D\to B$ and 2-cell $\sigma :fp\prime \Rightarrow gq\prime$ there is a 1-morphism $u:D\to (f/g)$ and isomorphisms $pu\cong p\prime$, $qu\cong q\prime$ such that modulo these isomorphisms, we have $\sigma u=\alpha$. There is also an additional “2-dimensional universality” saying that given $u:D\to (f/g)$ and $v:D\to (f/g)$ and 2-cells $\mu :pu\to pv$ and $\nu :qu\to qv$ such that $\alpha v.f\mu =g\nu .\alpha u$, there exists a unique 2-cell $\beta :u\to v$ such that $p\beta =\mu$ and $q\beta =\nu$. Note that the 2-dimensional property implies that in the 1-dimensional property, the 1-morphism $u$ is unique up to unique isomorphism. A square containing a 2-cell with this property is sometimes called a comma square.

A strict comma object is analogous but has the universal property of a strict 2-limit. This means that given $p\prime$, $q\prime$, and $\sigma$ as above, there exists a unique $u:D\to (f/g)$ such that $pu=p\prime$, $qu=q\prime$, and $\sigma u=\alpha$. Note that any strict comma object is a comma object, but the converse is not in general true.

In Cat, a comma category is a comma object (in fact a strict one, as normally defined); these give their name to the general notion.