I was reading the post Is there a name for when we extend the codomain of a function? (sort of like the opposite of the restriction) and have a similar question, however more simple and surely less interesting: What is the codomain of a restriction?
I mean, if I have the function $f:U\to V$ and $A\subset U$,
the restriction is the function $f|_A\to \square$ (I mantain the $V$? I've said a professor writing $f(A)$ as codomain, but so every restriction is surjective?)?
Many thanks.
The restriction of a function still has the same codomain. So, if $f:U\to V$ and $A\subseteq U$, then the restriction $f|_A$ is a function $A\to V$.
Of course, you can always shrink the codomain to any set which contains the image, and people usually don't bother to give the resulting function a separate name. So, if you take $f|_A$ and change its codomain to just $f(A)$ to get a function $A\to f(A)$, that function will commonly also be referred to as $f|_A$.