It is probably a trivial question. But I don't see the answer (and I didn't find anywhere).


Given a (complete and cocomplete) category X and an object A of X, we can define the "undercategory" A/X. See <http://ncatlab.org/nlab/show/under+category> 


I have already noticed that the coproduct of a set {i\_l: A to X} is the "natural injection" of A in the colimit of the obvious diagram defined by the set. 


I'm trying to understand how the product in A/X looks like, in terms of colimits, limits, products or coproducts of X. 


I appreciate any help. Thank you very much!



Product in undercategory
For a family of objects $A \to B\_i$ in $A/X$, their coproduct is usually called the [*pushout*](http://ncatlab.org/nlab/show/pushout) of the morphisms $A \to B\_i$ in $C$. It represents the functor $X \to \mathrm{Set}$, which maps $P$ to the set of families of morphisms $B\_i \to P$ which "coincide" on $A$. Pushouts in $X$ may be constructed via coproducts and coequalizers in $X$. The idea is to take $\coprod\_i B\_i$ and then identify $A \to B\_i$ with $A \to B\_j$.


The products in $A/X$ are more easy: The forgetful functor $A/X \to X$ preserves (also creates) them. This means that for a family of objects $f\_i : A \to B\_i$ the product is given by $f : A \to \prod\_i B\_i$, where $f$ is defined by $\mathrm{pr}\_i f = f\_i$.


