Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define
$$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if $x \in B$} \end{cases}. $$
Is there a standard name or notation for combining two maps $f,g$ into a map $h$ in this manner?
In set theory, at least, writing $f\cup g$ is perfectly acceptable and well understood. I don't think I ever heard a name for that, though.