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.

I will see if I can track down further citations.

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.

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.