Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i \hookrightarrow \prod_{i}A_i$?
Maybe a useful set of definitions is the following: Call a direct sum $\oplus_{i}A_i$ the image of $r:\coprod_i A_i\rightarrow \prod_i A_i$ given by
$$r_{ij}:A_i \rightarrow A_j, \;\; r_{ij}=1_{A_i}, i=j, \;\; r_{ij}=0, i\neq j.$$
(This is a definition taken from nLab.)
Then what I am actually interested in is the question
In a complete cocomplete Abelian category, is it true that $r$ is mono, i.e. $\coprod \simeq \bigoplus$?
I think I've read it on several places (e.g. planetmath, first sentence in "infinite products and coproducts"), but I am still suspicious, mainly because I don't see a reason why this should be true e.g. in categories of quasi-coherent sheaves. If this is not true, are there any hands-on examples, where this fails?
Thanks in advance for any help.