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What’s the notation for a function restricted to a subset of the codomain?

This is a really simple notation/terminology question, but I want to talk about it in a lecture tomorrow so want to be sure I'm using the right name.

If I have a function $f \colon X \to Y$ and a subset $Z \subset X$ then I can define the restriction of $f$ to $X$ and it can be written $f |_Z \colon Z \to Y$. What I want to know is whether or not there is an official name (or "widely accepted" name) and notation for when one restricts the function based on the target space. That is, when we have $Z \subseteq Y$ (containing $\operatorname{im} f$, of course) and restrict the codomain of $f$ to $Z$ instead of $Y$.

Nothing springs to mind, and it's hard to google something that you don't know the name of! (A few obvious ones didn't bring up anything.)

Answer 1

I usually call the corresponding subset of $X$ the inverse image of $Z$, denoted $f^{-1}(Z)$. But I can't recall seeing a standard notation (other than the obvious, but a bit over-complicated, $f|_{f^{-1}(Z)}$.