No arrows in pullbacks notation 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?
 A: The pullback is the categorical product in the slice category of objects over $C$. 
A nice motivating case to think about is the category of schemes, where, for example, the category of schemes over a field $k$ is the slice category of objects over $\text{Spec } k$ in the category of schemes. On affine schemes the categorical product is even given by tensor product of the corresponding rings of functions over $k$ (which note that we also denote by $A \otimes_k B$ without explicitly highlighting the dependence on maps $k \to A, k \to B$). 
Note that mathematical notation constantly omits dependence on extra data for the sake of brevity: for example, in representation theory it's typical to talk about a representation $V$ of a group $G$ without explicitly highlighting that all constructions involving $V$ depend on a choice of map $\rho : G \to \text{Aut}(V)$. The point is that it's understood that the symbol $V$ already implicitly refers to this data, which is usually clear from context. 
A: 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.
