TITLE: No arrows in pullbacks notation
QUESTION [8 upvotes]: Usually, in a category with pullbacks, the pullback on a cospan $f : A \rightarrow C \leftarrow B : g $ is noted $ A \times_C B $. However its definition is highly dependent on the choice of arrows $f$ and $g$.
So, is there a good coherent reason to mention $C$ in the notation, instead of mentionning $(f,g)$?
Or, if not, is there a good historical reason?

REPLY [4 votes]: You are technically correct. The definition of the pullback completely depends on the arrows $f$ and $g$. One of the reasons for the prevalence of the notation $A\times_C B$ instead of something like $A\times_{f,g} B$ is that it is more aesthetically pleasing and very often does not result in ambiguity. Unless you are in the presence of  another arrow with codomain $C$, the notation $A\times_C B$ can only refer to one pullback.