nLab
source

Definition

The source object, or simply source, of a morphism $f:x\to y$ in some category $C$ is the object $x$. The source of $f$ is also called its domain, although that can be confusing in categories of partial functions.

Given a small category $C$ with set of objects $C_0$ and set of morphisms $C_1$, the source function of $C$ is the function $s:C_1\to C_0$ that maps each morphism in $C_1$ to its source object in $C_0$.

Generalising this, given an internal category $C$ with object of objects $C_0$ and object of morphisms $C_1$, the source morphism of $C$ is the morphism $s:C_1\to C_0$ that is part of the definition of internal category.

Warning: there is another meaning of ‘source’ in category theory; see sink.