Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over a field $k$. An elementary finite correspondence from $X$ to $Y$ is defined to be an irreducible closed subset $W$ of $X\times Y$ whose associated integral subscheme $\widetilde{W}$ is finite and surjective over $X$. The group $Cor\_k(X,Y)$ of finite correspondences is the free abelian group on the elementary finite correspondences. My question is:



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> Why does one puts the conditions of being *finite* and *surjective* on the morphism $\widetilde{W}\to X$? 
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Given elementary finite correspondences $V$ and $W$ from $X$ to $Y$ and $Y$ to $Z$ respectively, it is shown in [Mazza-Voevodsky-Weibel](http://fermat.dima.unige.it/~mazza/MVW.pdf) Lemma 1.7 that $V\times Z$ and $X\times W$ intersect properly and also $\widetilde{V}\times\_Y\widetilde{W}$ is finite and surjective over $X$, so that composition of correspondences can be defined. 



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> Is this statement true in the reverse direction? Is this the only reason fro the choice of the definition of finite correspondences?
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Intuition behind the definition of finite correspondences
Traditionally correspondences were defined simply as cycles on the product, but then you need a moving lemma just to define composition. This limits you to working on smooth varieties. The beauty of finite correspondences is that composition can be defined much more directly, essentially as a composition of multivalued functions (cf Denis Nardin's comment). So by limiting the kind of correspondences you allow, you increase the range applicability. 


