How can I see that the functor
$\displaystyle
A \mapsto A \otimes_{\mathbb Z} A
$
from the category of abelian groups to itself is not additive.
if it was additive, then we'd have: $\displaystyle f(x) \otimes g(y) + g(x) \otimes f(y) = 0,$ for all $\displaystyle x,y \in G_1,$ all abelian groups $\displaystyle G_1, G_2,$ and all $\displaystyle f, g \in \text{Hom}_{\mathbb{Z}}(G_1,G_2).$ but this is nonsense! (consider trivial examples!)