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.