Composing functions

Question

Is there any common notation for a composed version of functions $F:X\to Y$, $G:X\to Z$ given as $H(x)=(F(x),G(x))$ and of functions $F:X\to Z$, $G:Y\to W$, given as $H(x,y)=(F(x),G(y))$?

Answer 1

I would write $( f, g )$ for $x \mapsto (f(x), g(x))$ and $f \times g$ for $(x, y) \mapsto (f(x), g(x))$. The notation $(f, g)$ reflects the fact that $x \mapsto (f(x), g(x))$ provides the witnesses showing that the bifunctor $(\times):(X, Y) \mapsto X \times Y$ is the product in the category of sets of the sets $X$ and $Y$. The notation $f \times g$ reflects the fact that $(x, y) \mapsto (f(x), g(x))$ gives the morphisms part of that bifunctor.