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.

tuplingof the list $f_1, \ldots, f_n$. $\endgroup$ – Todd Trimble♦ Oct 5 '15 at 21:54General topology, which is very precise about terminology. It uses the term "diagonal of mappings" and the notation $\bigtriangleup_{i=1}^n f_i$. (the definition is at the bottom of page 79 in the 1989 edition) $\endgroup$ – Niels J. Diepeveen Oct 6 '15 at 0:16cartesianproduct I use the same name for $f=(f_1,\ldots,f_n)$. $\endgroup$ – Jochen Wengenroth Oct 6 '15 at 6:27