New symbol for an exact sequence A sequence $A\to B \to C$ is exact if $\operatorname{im} f = \ker g$, where $ f:A\to B$ and $g:B\to C$.

Why is there not a symbol to denote such a sequence?
Which one would you suggest?
@egreg's comment is totally right — because there's no need in such symbol. 

Also, there are at least two reasons to not introduce it. First one is that heavily symbolized mathematical writing is usually unreadable. Second is high probability of  ambiguity in case when diagram is somewhat complicated. 

So I would suggest to just write «this sequence is exact at $B$», or that «pair of morphisms $f, g$ is exact» (which is better, in my opinion).
Here are three ideas. I prefer the last one.