nLab
restriction

The restriction of a function $f:X\to Y$ to a subset $U$ of $X$ is simply the composite of $f$ and the inclusion function? of $U$:

$$U\stackrel{i_U}{\to }X\stackrel{f}{\to }Y.$$U \stackrel{i_U}\to X \stackrel{f}\to Y .

It is often written as $f_{\mid U}:U\to Y$, or a variation on that theme.