Does this covariant functor really turn coproducts into products?

Let $X$ be a set. Let $\mathrm{Over}(X)$ be the category of sets $A$ with a map $A\to X$; the morphisms are equivariant maps, that is, maps which make the obvious triangle commute.

A map of sets $f\colon X\to Y$ induces a functor $f_*\colon\mathrm{Over}(X)\to\mathrm{Over}(Y)$ by postcomposition. This shows that $\mathrm{Over}$ is a functor from the category of sets to the category of categories (up to usual set theoretic issues which I would like to ignore).

Now let $X=X_1\sqcup X_2$ be a disjoint union, i.e., a coproduct of sets. Every set $\phi\colon A\to X$ gives a pair consisting of a set over $X_1$ and a set over $X_2$. These are the restrictions $\phi^{-1}(X_i)\to X_i$. On the other hand, if you give me a set over $X_1$ and a set over $X_2$, then I can write down a corresponding set over $X$ by just taking the disjoint union.

I think what I just said means that the coproduct decomposition $X=X_1\coprod X_2$ gives a product decomposition of categories $\mathrm{Over}(X)\cong \mathrm{Over}(X_1)\times \mathrm{Over}(X_2)$.

I am confused by the apparent fact that the covariant functor $\mathrm{Over}$ turns coproducts into products. What is happening here?

Edit: As Theo explains in his comment below, the coproduct of categories is just the disjoint union. So it is not a very complicated thing, but also not what we want here---which is really pairs of objects.

Edit: Does the pullback diagram $$\begin{array}{ccc} P & \rightarrow & A\\ \downarrow_{f^!(\phi)} & & \downarrow_\phi\\ X & \rightarrow^f & Y \end{array}$$ define a wrong-way functoriality for maps $f\colon X\to Y$ which would turn the canonical maps of a coproduct $X_1\sqcup X_2$ to the canonical maps of the product $\mathrm{Over}(X_1)\times \mathrm{Over}(X_2)$? This seems to correspond to Omar's observation.

• A first remark: The coproduct of categories is simply the disjoint union. The free product of groups (I guess that's what you have in mind) is so complicated because one insists that there be only one object. I haven't thought about the question yet. – t.b. Mar 24 '11 at 10:50
• @Rasmus: What does $\phi^{-1}(X)$ in “Edit” mean? $X$ is not a subset of $Y$. – beroal Mar 24 '11 at 13:27
• @beroal: you are right, thanks. – Rasmus Mar 24 '11 at 15:22
• @Rasmus: Pullbacks in fact define the pullback (base change) functor $f^*$, see ncatlab.org/nlab/show/base+change . Your $f_*$ is $f_!$ there, also $\Sigma_f$ in other texts. It seems that $-^*$ turns coproducts into products, though I doubt that this is true after replacing Set with an arbitrary category. Then you need to prove $\forall i\in \{0,1\}. \iota_i^*(\coprod_j \phi_j)\cong \phi_i$, $\coprod_i (\iota_i^*(\phi))\cong \phi$. This material is too advanced for me, I suggest you to ask on mathoverflow.net . – beroal Mar 25 '11 at 4:14
• I wouldn't recommend asking this on mathoverflow, they'll probably just send you back here for this. – Omar Antolín-Camarena Mar 25 '11 at 12:47