Abstract Algebra – The name "section" for the selection of representatives of an equivalence class

This is a question about terminology and sources. Looking for a name for the operation "Selecting an Equivalent Representative" I came across the Wikipedia article on equivalence classes, which calls this operation a "section."

This term is given an equivalence relationship $ sim $ on a sentence $ X $Can I define a "canonical section" $ s_c $ as an injective card $ s_c colon X / sim to X $ and then add the picture $ s_c $ An equivalence class is called a "canonical representative". This notation is very useful for formal derivations that I can use $ s_c (C) $ instead of natural language descriptions like "and then we take the canonical representative of C, …".

However, this use of the term "section" does not seem to be common in textbooks on algebra, so I'm not sure I should use it in my current research. I think it comes from using "section" and "withdrawal" in category theory, but would it be understood by people with a background in other areas of mathematics? Is there a well-known textbook that deals with equivalence relationships, where the word "section" is used in this way?