Coproducts and products of modules I've just looked at the book A Course in Homological Algebra (by Hilton and Stammbach). They show the universal property of the direct sum (coproducts) using injections and the universal property of the direct product using projections. Then they prove that $$\text{Hom}( \coprod_i A_i, B) \cong \prod_i \text{Hom} (A_i, B)$$ and $$\text{Hom}(A, \prod_i B_i) \cong \prod_i \text{Hom} (A, B_i).$$ 

The question is why not use injections in the universal property of the coproduct of modules and why not $\text{Hom}( \prod_i A_i, B) \cong \prod_i \text{Hom} (A_i, B)$? 

I know it differs for general categories, because of the duality, but, for modules, isn't it possible to do this kind of generalization? 
Thanks in advance.
 A: It's a Goldilocks situation as the canonical injections and projections are just the right thing to prove the claims at hands. Moreover, direct sums and direct products also differ for modules.
Consider the case $B=A_i=\mathbb Z$ with index set $\mathbb N$.
Then $\prod_{i} A_i$ is the set $\mathbb Z^{\mathbb N}$ of functions $\mathbb N\to \mathbb Z$.
What is $\prod_i \operatorname{Hom}(A_i,B)$? An element of this is determined by specifying for each $i\in \mathbb N$ an element of $B$ (the image of $1\in A_i$), hence "is" also $\mathbb Z^{\mathbb N}$ and countable.
What is $\operatorname{Hom}(\prod_i A_i,B)$? It is definitely bigger than $\mathbb Z^{\mathbb N}$ already because $\prod A_i$ is uncountable!
A: 
Why not  $\text{Hom}( \prod_i A_i, B) \cong \prod_i \text{Hom} (A_i, B)$ ? 

$\text{Hom}( \prod_\mathbb{N} \mathbb{Z}, \mathbb{Z}) \cong \bigoplus_\mathbb{N}\mathbb{Z}$ is a free abelian group while $\prod_\mathbb{N} \text{Hom} (\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}^\mathbb{N}$ isn't free. Hence they are not isomorphic. 
As reference for the first isomorphism see https://mathoverflow.net/questions/105771/is-the-dual-of-the-product-of-infinite-cyclic-groups-a-free-abelian-group and http://www.math.uni-duesseldorf.de/~schroeer/publications_pdf/infinite_product-1.pdf for the latter. 
