Q:

Are the natural injections always injections in any concrete category?

I am beginning to learn about categories in Aluffi's Algebra: Chapter 0. 
In every example I know, the coproduct's natural injections are always injections. Does this happen in any concrete category? If so, in general categories are they always monomorphisms?
Since I don't know a lot about categories yet, I appreciate keeping the language simple. Thanks!

A:

It is not true in general: Consider the concrete category of commutative rings, where coproducts are given by tensoring over $\mathbb Z$. Now consider $\mathbb Z\to \mathbb Z\otimes \mathbb Z/2\mathbb Z=\mathbb Z/2\mathbb Z$, which is clearly not injective.

