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:

Why does one puts the conditions of being

finiteandsurjectiveon the morphism $\widetilde{W}\to X$?

Given elementary finite correspondences $V$ and $W$ from $X$ to $Y$ and $Y$ to $Z$ respectively, it is shown in Mazza-Voevodsky-Weibel 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.

Is this statement true in the reverse direction? Is this the only reason fro the choice of the definition of finite correspondences?