An example about finitely cogenerated modules We know the fact that: If a module $M$ is finitely cogenerated, then every module that cogenerates $M$ finitely cogenerates $M$. Conversely, it is not true.  I find an example in the book "Rings and Categories of modules" written by Frank W. Anderson and Kent R. Fuller. 
Example The abelian group $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$, ($p$ is a prime) is not finitely cogenerated yet every group that cogenerates it finitely cogenerates it.
I am at a loss for this example. Any help will be appreciated. Clearly, we can regard the abelian group $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$ as a module over $\mathbb Z$. 
I post my effort. (1) $\mathbb{Z}_p$ is simple. (2) $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$ is semisimple and  since $\mathbb{P}$ can be $\infty$,  $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$ is not finitely generated, then it is not finitely cogenerated.
 A: I think what you've written so far is in the right direction, but probably could be expressed more directly this way:
$\bigoplus_{p\in P}\mathbb Z_p\hookrightarrow \prod_{p\in P}\mathbb Z_p$ obviously cannot be pruned down to a finite subset $F\subseteq P$, so $\bigoplus_{p\in P}\mathbb Z_p$ is not finitely cogenerated. (Eliminating the position in the product for $p$ would remove all possible nonzero images for an element of order $p$ in $\bigoplus_{p\in P}\mathbb Z_p$.)
Then for the reverse direction, the point is to establish that if $\bigoplus_{p\in P}\mathbb Z_p\hookrightarrow\prod_{i\in I}G$, you can embed it into finitely many copies of $G$. In this case, actually, I think it's true that $\bigoplus_{p\in P}\mathbb Z_p\hookrightarrow G$.
Let $x_p$ generate $\mathbb Z_p$ and $\phi$ be the embedding. $\phi(x_p)$ is nonzero on some position in $\prod _{i\in I}G$, and we can project on that coordinate to get a nonzero element of $G$, necessarily of order $p$. So $G$ contains a copy of $\mathbb Z_p$. This is true for every prime $p$, and obviously owing to the prime orders of these copies, their sum is direct. So $G$ contains a copy of $\bigoplus_{p\in P}\mathbb Z_p$.
