Question about Algebraic Spaces I have two questions about Knutson's Algebraic Spaces.


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*Can one clarify Prop 5.2 to me which says the following. Let $R\rightarrow U\times U$ be a categorical equivalence relation. Then there is a unique map $i:U\rightarrow R$ such that the composite $U\rightarrow R\rightarrow U\times U$ is the diagonal.

*Can one clarify the second line of the proof of Prop 5.9 starting with "Then"? Assume we have maps of sheaves $V^{\cdot}\rightarrow U^{\cdot}\rightarrow F$. Why is it true that
$V^{\cdot}\times_{F}U^{\cdot}=V^{\cdot}\times_{U^{\cdot}}U^{\cdot}\times U^{\cdot}=V^{\cdot}\times_{U^{\cdot}}R^{\cdot}$?
 A: It may help you (in understanding these and related questions) to apply Yoneda's lemma bit more conciously in your thinking.  Yoneda's lemma is pretty easy as a technical result, but incredibly powerful as a psychological tool!
For example, to say that $R$ is a categorical equivalence relation on $U$ is (unless I've got the definition wrong) to say that for any $X$ in the category,
the map $$Hom(X,R) \to Hom(X,U) \times Hom(X,U)$$ induced by the given map $R\to U\times U$ induces an equivalence relation on $Hom(X,U)$.
In particular, the diagonal $Hom(X,U)$ must be contained in $Hom(X,R)$.  This holds
for all $X$, so by Yoneda, the diagonal $U \to U\times U$ factors through $R$.
For the second point, I can't quite work out the context from what you've written, but it should follow from universal properties of fibre products (such as $(X \times_Z Y) \times _Y W = X \times_Z W\,$), and again Yoneda should help, i.e. you just have to check that the two expressions represent the same functor.
E.g. here is how you could verify the identity I wrote above:
Giving a map (from some $T$, say) to $(X \times_Z Y) \times_Y W$ is the same as giving maps
to $(X\times_Z Y)$ and to $W$ lying over the same map to $Y$, which is the
same as giving maps to $X$ and $Y$ lying over the same map to $Z$, and a map to $W$ lying over the given map to $Y$.
Since the map to $Y$ is determined by the map to $W$, we can forget it, and just think about the maps to $Z$; thus we just have to give a map to $X$ and $W$ lying over the same map to $W$.  
(This is easier if you write down all the relevant maps with pen and paper!)

Here is something more specific about question (2):
First note that $R$ is equal to the fibre product $U \times_F U$ (it gives the equivalence relation modulo which morphisms to $U$ are being identified to give
rise to the quotient sheaf $F$).
So we get that $$V \times_F U = V \times_U (U\times_F U) = V\times_U R;$$
the first identification is the one already discussed in what I wrote earlier,
and the second comes from the fact I just mentioned.
