How to find a direct product in a category?

Question

Having a category how to find a direct product in this category?

Is it entirely a guesswork (try this function, try that one) or is there a method for this?

Answer 1

Assuming that by 'direct product' you mean 'categorical product': If the category is totally arbitrary, then yes, it's totally guess work. Of course, most often the category in question is far from arbitrary. The category could belong to a familiar type of categories. For instance, concrete categories, categories of algebras of a monad, categories of algebras of an operads, categories of models for a certain type of theory, and so on. For each of these families of categories there is often a common way to construct (co)limits of a certain shape.

Very often too, a category $C$ is related to some other category $D$ by some functor $F:C\to D$. If that functor is a right adjoint and products are known in $D$, then $F$ would preserve products, and thus finding products in $C$ is less of a guess work in that case. (See the relevant sections in Mac Lanes's Categories for the working mathematician, discussing functors reflecting/creating (co)limits).

Also, in the presence of an object $*$ in $C$ for which $Hom(*,X)$ characterizes, in some way, the elements in $X$ (think for instance of categories like $Set$, where $*$ is a singleton, but not of categories like $Grp$), then since $Hom(*,A\prod B)$ is, by definition, naturally isomorphic to $Hom(*,A)\times Hom(*,B)$, less guess work is needed.