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In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others).

But only one of these products ($A\times B$) is considered "canonical". Is there any formal or informal term about the "main" of the products?

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  • $\begingroup$ Just to be sure you know this: The direct product $A \Pi B$ of some objects $A$ and $B$ in some category always comes with some arrows $A \Pi B → A$ and $A \Pi B → B$, so the direct product consists of three informations. Then, of all the isomorphic direct products, in $\mathbf{Set}$ choosing $A ×B$ with the two coordinate projections is, of course, an obvious choice of a direct product. One can then argue that $A×B$ can be considered somehow canonical, since the symbols “$A×B$” already implicitly give the associated arrows to $A$ and $B$, informally speaking. $\endgroup$ – k.stm Sep 9 '13 at 13:56
  • $\begingroup$ It's also then true that such a choice of three pieces of information is unique up to unique isomorphism (once "isomorphism" is defined appropriately), so I'm not sure it's really necessary to single out one choice as canonical. $\endgroup$ – mdp Sep 9 '13 at 14:10
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    $\begingroup$ Actually it is not even possible to define $A\times B$ as more canonical than $B\times A$ since the product is defined as a universal cone over the discrete diagram of two objects, and this diagram has no specified order. $\endgroup$ – Stefan Hamcke Sep 9 '13 at 14:17
  • $\begingroup$ I’m curious: Why is this question downvoted? $\endgroup$ – k.stm Sep 9 '13 at 15:21
  • $\begingroup$ I have downvoted since the assumption of the question is wrong. There is no canonical product, as explained by Matt Pressland. Also the question itsself is ill-posed. $\endgroup$ – Martin Brandenburg Sep 9 '13 at 16:36
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There is no point in singling out a specific product in general category theory, because all (non-evil) properties of a specific product hold for any product. In concrete examples this means you're free to choose the most convenient product for the task at hand.

Having additional structure like a partial order on the set of objects or morphisms doesn't change that fact, because the categorical product - as it is defined for general categories - simply doesn't "see" that structure (unless there is some deeper connection between the structure of the category and the given partial order). If you add structure to an existing algebraic object, you might wanna tinker the existing structure to respect the new one. E.g. there is the notion of a locally partially ordered category, which imposes the condition on morphisms to satisfy $f_1\le f_2 \wedge g_1\le g_2\implies g_1\circ f_1\le g_2\circ f_2$. But then again you won't find a general canonical choice of a product.

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