1. ## Category

How can I see that the functor

$

A \mapsto A \otimes_{\mathbb Z} A
$

from the category of abelian groups to itself is not additive.

2. Originally Posted by peteryellow
How can I see that the functor

$

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: $f(x) \otimes g(y) + g(x) \otimes f(y) = 0,$ for all $x,y \in G_1,$ all abelian groups $G_1, G_2,$ and all $f, g \in \text{Hom}_{\mathbb{Z}}(G_1,G_2).$ but this is nonsense! (consider trivial examples!)