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For every product in a category, there exists an operation $\langle\cdot, \cdot\rangle$ that turns morphisms $f : C \to A$ and $g : C \to B$ into morphisms $\langle f, g\rangle : C \to A \times B$. How are this operation and its results usually called? Calling $\langle f, g\rangle$ the product of $f$ and $g$ does not make sense to me, since I would expect $f \times g$ to be the product of the two morphisms.

Furthermore there is the generalisation of binary products in the form of arbitrary products $\prod_{i \in I} A_i$ for families $\{A_i\}_{i \in I}$ of objects. These products have, of course, variants of the $\langle\cdot, \cdot\rangle$-operator. What is the usual notation for these variants? So far, I wrote $\langle\{f_i\}_{i \in I}\rangle : C \to \prod_{i \in I} A_i$ for families of morphisms $f_i : C \to A_i$. Is this standard?

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Maybe $f \times_C g$? –  Qiaochu Yuan Aug 11 '12 at 17:43
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up vote 7 down vote accepted

Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the operation have called it the "pairing arrow."

I would write $\langle f_i\rangle_{i \in I} : C \to \prod_{i \in I} A_i$ without curly brackets in the angle brackets and i would call it the arrow to the product. A more specific name for the operation could be "tupling arrow."

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+1 for the "pairing" of $f$ and $g$; I think this is the most common term and the most sensible. I've seen it written $(f,g)$ as well as $\langle f,g\rangle$; either seems fine to me. –  Mike Shulman Aug 13 '12 at 6:02
    
By the way, how would you call the corresponding operaton $[\cdot, \cdot]$ for coproducts then? –  Wolfgang Jeltsch Aug 13 '12 at 8:30
    
@Colin: The notation $\langle f_i\rangle_{i \in I}$ looks nice and sensible indeed. –  Wolfgang Jeltsch Aug 13 '12 at 8:31
    
@Wolfgang: "cotupling" has been used. I suppose in Colin's scheme that would become "copairing"? –  Chris Heunen Aug 13 '12 at 9:27
    
I've most often heard "copairing". –  Mike Shulman Aug 15 '12 at 3:20
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