Fiber product of varieties vs schemes reference Given two complex varieties over a common base, I can take their fiber product in the category of varieties, or I can take their fiber product in the category of schemes and then take the reduced subscheme. I have heard, that these two operations yield the same.
Has someone a reference? 
 A: Here is a proof of this fact:
Let $\mathcal C$ be the category of (not necessarily irreducible) complex varieties.
Then $\mathcal C$ can be identified with the category of reduced finite type $\mathbb C$-schemes.
Let $\mathcal D$ be the category of all finite type $\mathbb C$-schemes.  
Then obviously $\mathcal C$ is a full subcategory of $\mathcal D$, and the inclusion
$\mathcal C \subset \mathcal D$ has a right adjoint, namely passage to the underlying reduced subscheme.  General nonsense (i.e. an easy categorical argument) then shows that if $X\to S$ and $Y \to S$ in $\mathcal C$ are two morphisms, the fibre product in $\mathcal C$
can be computed by first computing the fibre product in the bigger category $\mathcal D$,
and then applying the right adjoint to the inclusion.  That is,
the fibre product in the category of varieties over $\mathcal C$ is equal to
the reduced subscheme of the fibre product in $\mathcal D$ (which coincides with
the fibre product in the category of all schemes, just because the fibre product
of morphisms of finite type $\mathbb C$-schemes is again finite type over $\mathbb C$).
I'm not sure of a reference.  Because the proof is easy when you have the right
categorical framework, it is the kind of thing that is well-known to experts but whose proof
is not necessarily written down explicitly.
