Let $g(x,y)$ be a function defined on $\mathbb{R}^2$, such that $g$ is decreasing with respect to $x$ and decreasing with respect to $y$. and let $f(x,y)$ be a function defined on $\mathbb{R}^2$, such that $f$ is decreasing with respect to $x$ and decreasing with respect to $y$
what is the monocity of $f(g(x,y),y)$ with respt to $y$ ?
Can we deduce that $f(g(x,y),y)$ is increasing with respect to $y$ as a composition of two decreasing functions?
I don't think so :
take $g(x,y)=-x-y$ and $f(x,y)=-x - 2y$, then you have $$f(g(x,y),y)=x+y-2y=x-y$$ which is decreasing with respect to $y$.