Is there a standard notation for the "two-sided slice"?

This is: given $f\colon A\star B\to C$, the object which represents the functor $S\mapsto$ set of maps $g\colon A\star S\star B\to C$ such that $g|A\star \varnothing\star B=f$.

@CharlesRezk Nope. I don't think it's a widely-considered notion in classical category theory. It's definitely a weird construction, kind of like a "constricted comma" or like a "category of bicones"

if you have to choose a symbol, maybe an hourglass-shaped symbol might be cute?