Let $A,B,X,Y$ be $4$ sets. Let $f:X\to A$ and $g:Y\to B$ be a pair of functions. What is the common definition of the expression $\arg\min\{f(x),g(y)\}$?
As noted by the comment, $\text{argmin} f(x)$ is the $x$ that gives smallest $f$. But what happens when $\text{argmin}$ takes two function, what does say?