TITLE: Notation for a function from disjoint union
QUESTION [9 upvotes]: Given two functions $f:A\rightarrow C,\ g:B\rightarrow C$, is there a standard way to denote the function $h: A \sqcup B \rightarrow C$ such that $h\mid_A\equiv f$ and $h\mid_B\equiv g$? Is it something like $f\amalg g$ or $f\sqcup g$?

REPLY [6 votes]: You can use the same notation as for sets. Not only because "everybody will know what you mean", but also because it is coincidentally well defined. This is because functions are sets of tuples (the set theoretic definition of functions). If you regard the functions $f$ and $g$ as sets, then the set $f\sqcup g$ is again a function and it is exactly the function you wanted to get. Indeed:
Writing $f$ and $g$ as sets yields
$$f = \{(a,y)\mid a\in A \mbox{ and } y = f(a)\}$$
$$g = \{(b,y)\mid b\in B \mbox{ and } y = f(b)\}$$
Taking the ordinary disjoint union of sets yields
$$f\sqcup g = \{(c,y)\mid (c\in A \mbox{ and } y = f(c)) \mbox{ or } (c\in B \mbox{ and } y = f(c))\}$$
which agrees with your definition of disjoint union of functions.