If $f : X \to Y$ and $g : A \to B$ are functions, we can define several products on them:
1) If $X = A$ and $Y = B$ and we have some kind of product on $Y$, we can define the pointwise product $(f \cdot g) : X \to Y$ as $$ (f\cdot g)(x) = f(x)\cdot g(x) $$ (this is also the product often involved when building the direct product of algebraic objects).
2) In general, we can define the cartesian product of functions $(f\times g) : X \times A \to Y \times B$ by $$ (f\times g)(x,a) := (f(x), g(a)). $$
3) If $X = A$ we can define $(f, g) : X \to Y \times B$ by $$ (f, g)(x) := (f(x), g(x)). $$
I saw the third construction used implicitly in many textbooks for example when referring to closure properties under composition. But seldom I see this construction written out? So has it any name?
Surely if $\pi_1 : X \times A \to X, \pi_2 : X \times A \to A$ are the projection mappings we have $$ (f\times g) = (f\circ \pi_1, g \circ \pi_2) $$ and if $X = A$ and $\Delta : X \to X \times X$ denotes the diagonal inclusion $\Delta(x) := (x,x)$ we have $$ (f,g) = (f\times g) \circ \Delta. $$ And also if we suppose all involved sets come from some universal set $U$ closed under cartesian product and assume $Y,B \subseteq U$, hence assume $f : X \to U, g : A \to U$, we have that $(f,g)$ is the pointwise product w.r.t. the cartesian product in $U$, but this seems quite artifical to me. Furthermore if $X = A$ and $Y = B$ and $\cdot : Y \times Y \to Y$ denote the product on $Y$ we have $(f\cdot g) = \cdot \circ (f,g)$.
These are all the relations between the product that come to my mind. But anyway, is there any name for the third one. And is it under consideration in some areas of mathematics, or explicitly mentioned? Or is this just to trivial?