nLab
inclusion function

Given a set $X$ and a subset $A$ of $X$, the inclusion function of $A$ in $X$ is the function $i_A:A\to X$ given by

for every element $n$ of $X$ that belongs to $A$.

Every inclusion function is an injection, and every injection is isomorphic (in the slice category $\mathrm{Set}/X$) to an inclusion function. In appropriate categories, it is common to analogously interpret monomorphisms or regular monomorphisms as inclusions of subobjects. One then speaks of inclusion morphisms.

The inclusion function of $A$ in $X$ is the restriction to $A$ of the identity function on $X$. Conversely, the restriction to $A$ of any function $f:X\to Y$ is the composite of $f$ after the inclusion function: