i've just gotten into the theory of motives.I understand the construction of the Karoubian envelope (pseudo-abelian completion) to ensure that morphisms have kernels and images in order to get certain decompositions of the diagonal of Motives in $\mathcal{M}_k$.But i don't see why the morphisms in $Corr_k$ which are elements of $CH(X \times Y)$ for the objects $X,Y$ don't have kernels and/or images in general.

A second one: In the category of effective Chow-Motives, do you know an example of non isomorphic varieties,that have the same motive,except for a totally split quadric and projective spaces?