Def/Descent of functions

 Often, the process of "descent of functions" is implicitly stated with the phrase "well-defined". Instead of saying that "$f$ descends to a function $\bar f \colon \bar X \rightarrow Y$", one might say that "$f$ yields a defines::well-defined function from $\bar X$ to $Y$". In this situation, one often uses the same letter or symbol $f$, for the unique function $\bar f$ from $\bar X$ to $Y$.