This is an embarrassingly simple question, but I was not able to find a definitive answer from literature search.


Suppose one has some collection of functions $f\_1: X \to Y\_1, \dots, f\_n: X \to Y\_n$ on a common domain $X$. Then one can form the function $(f\_1,\dots,f\_n): X \to Y\_1 \times \dots \times Y\_n$ in the usual fashion:


$$ (f\_1,\dots,f\_n)(x) := (f\_1(x),\dots,f\_n(x)).$$


My question is: what does one call the function $(f\_1,\dots,f\_n)$? I had (without thinking much about it) used to call it the "direct sum" of $f\_1,\dots,f\_n$, before realising that this actually had no relation with the usual meaning of direct sum. In category theory, one might call $(f\_1,\dots,f\_n)$ the "product" of $f\_1,\dots,f\_n$, but this could get confusing if $f\_1,\dots,f\_n$ already take values in some ring, so that there is also a pointwise product. Is there some other commonly accepted term for describing the function $(f\_1,\dots,f\_n)$? It doesn't sound quite right grammatically to refer to it as the "tuple" of $f\_1,\dots,f\_n$, and "concatenation" or "join" don't quite seem to fit either.



What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?
I was encouraged to make my comment an answer: 


In the case $n = 2$, I would call it the pairing. Similarly, one has "tripling", "quadrupling", and so in general one might call it the ($n$-)*tupling* of the list $f\_1, \ldots, f\_n$. And indeed that is what the nLab calls it: see [here](http://ncatlab.org/nlab/show/pairing). 


Whatever this should be called, I would *not* call it the cartesian product of $f\_1, \ldots, f\_n$. The product is a functor $\mathcal{C}^n \to \mathcal{C}$ whose value at a morphism $(f\_1: X \to Y\_1, \ldots, f\_n: X \to Y\_n)$ of $\mathcal{C}^n$ is rather the morphism $f\_1 \times \ldots \times f\_n: X \times \ldots \times X \to Y\_1 \times \ldots \times Y\_n$ of $\mathcal{C}$, and it's the latter that I would call the product of $f\_1, \ldots, f\_n$. 


I will see if I can track down further citations for $n$-tupling. 


