Say that $f: A \rightarrow B$ is an arrow in a category $\mathcal C$. Which verbs or expressions do we use to express in an alternative way that $A$ is the source of $f$ and $B$ its target? E.g., [does it make sense, is it usual] to say something like "$f$ springs out of $A$", or "$f$ [leaves, stems] from $A$", or "$f$ [lands into, arrives in] $B$"? If not, what are some correct, common expressions?
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I don't think it's very common to use a verb to describe $f$ without mentioning both $A$ and $B$. If you only want to mention one, then I have no great suggestions other than replacing source/target with domain/codomain. You might say "$f$ is defined on $A$" or "$f$ takes values in $B$," but that language is best for concrete categories. ``$f$ maps $A$ into $B$" is rather common, if you mention both objects. All of your suggestions were unfamiliar to me.