Is there an established name for this kind of function composition? $$ f : A → B $$ $$ g : A×B → C $$ $$ Cont(g, f)(x) = g(x, f(x)) $$ Is there an established name for $Cont$ operation?

$Cont(g,f)=g\circ (\rm Id_X\times f)$. So it is not just similar to function composition, it _is_ function composition.