$inf_Ysup_X f(x,y)$

Let $Y=(-1,0)$ and $X=(0,1)$ and $f(x,y)=2x+2y$

Let’s start simply with $inf_Y(sup(2x+2y:0<x<1))$

now here occurs the first problem for me, can I use the properties of $Y$ in this step, or would it be $inf_Y(2+2y)$ or since i know that Y is negative i could finish here and say $inf_Y(2)=2$

but the first approach would tell me it is: $inf_Y(2+2y)=2-2=0$ since $Y=(-1,0) =>inf_Y(2y)=-2$

So do I have to take sup of f(x,y) regardless of what Y is, which doesn’t make sense. Maybe i need a proper definition for multivaraible sup and inf

Please give a comprehensive answer which approach is correct. Any explanation is appreciated