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What is the simplest way to check whether a given function of two arguments (Its arguments and the value are morphisms of some category.) is a direct product in categorical sense?

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    @Zev Chonoles: In fact I have a function which takes two ARBITRARY (with arbitrary domains and codomains) morphisms of certain category. We may restrict this function to take only morphisms with identical domains. I suspect (after this restriction) it will be a direct product in categorical sense.2012-01-26

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You seem to be asking whether it is possible to give an essentially algebraic axiomatisation of categorical products. The short answer is: yes, but you need some additional data.

Let $\mathcal{C}$ be a category. Suppose we have the following operations:

  • For every pair of objects $(A, B)$, another object $A \times B$ and two arrows $\pi_{A,B} : A \times B \to A$, $\pi'_{A,B} : A \times B \to B$.
  • For every triple of objects $(A, B, C)$ and pair of arrows $f : C \to A$, $g : C \to B$, an arrow $\langle f, g \rangle : C \to A \times B$, such that the following axioms hold:

    1. $\pi_{A,B} \circ \langle f, g \rangle = f$

    2. $\pi'_{A,B} \circ \langle f, g \rangle = g$

    3. For all $h : C \to A \times B$, $\langle \pi_{A,B} \circ h, \pi'_{A,B} \circ h \rangle = h$

Exercise. Verify that the triple $(A \times B, \pi_{A, B}, \pi'_{A, B})$ has the universal property of the product of $A$ and $B$.

Some other universal constructions in categories can also be made essentially algebraic: this is done in the first chapter of Lambek and Scott's Introduction to higher order categorical logic, for example.

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    I$f$ you are given the operation (A,B)↦A×B, then a product preserving functor between categories with finite products must preserve the specified product. If you require only that there be an object A times B with the required properties, then a product preserving functor must take a product to SOME product of the images of A and B (of course, they are all naturally isomorphic). Note: This is a fine point that hardly ever makes any difference in practice! See Toposes, Triples and Theories at www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf, page 141.2012-07-24