# What is the notion for this arrow in category theory?

When reading a CS paper, I encountered an arrow like $\langle f_1, \ldots, f_n \rangle : \langle A_1, \ldots, A_n \rangle \rightarrow \langle B_1, \ldots, B_n \rangle$. The author took it for granted to mean a collection of arrows $f_1 : A_1 \rightarrow B_1$, ..., $f_n : A_n \rightarrow B_n$. I wonder if there is such a notion in category theory. If it is, I would appreciate a reference. Thanks.

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As far as I can tell, you can regard it as being an arrow in the product category $C\times C\times\dots\times C$. – Karl Kronenfeld May 3 '13 at 13:56
Ah, of course, thanks. Unfortunately you did not put it into an answer. – day May 3 '13 at 14:01
There is not enough context to determine what your notation means. It would be helpful if you could give a precise reference. – Zhen Lin May 3 '13 at 15:14
@user1 Please promote your comment to an answer, as requested by OP. – Lord_Farin May 23 '13 at 10:58
In category theory, Cartesian products are bifunctors $\mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$. So not only can we form the product $X \times Y$ for arbitrary objects $X$ and $Y$, but also the product $f \times g$ for arbitrary morphisms $f$ and $g$. Moreover, if $f_1 : X_1 \rightarrow Y_1$ and $f_2 : X_2 \rightarrow Y_2,$ then $f_1 \times f_2 : X_1 \times X_2 \rightarrow Y_1 \times Y_2$. – goblin Feb 16 '14 at 1:35

Based on the information provided, the arrow $\langle f_1,\dots,f_n\rangle$ can be regarded as an arrow in the product category $\mathscr D=\mathscr C\times\mathscr C\times\dots\times\mathscr C$ if each arrow $f_i$ originates in $\mathscr C$.