Interchanging Hom and direct sum---the example given by Mike Haskel I don't quite understand this example given by Mike Haskel. I want to find an example about
$$\operatorname{Hom}_R\left ( M ,\bigoplus_{i\in I} N_{i}\right )\not \cong\bigoplus_{i\in I} \operatorname{Hom}_R\left ( M ,N_{i}\right ).$$
Mike Haskel's example is that:

It's not true when $I$ is infinite, due exactly to the problem you encounter. Consider the case where $R = \mathbb{R}$, $M$ is an infinite dimensional vector space, and each $N_i$ is $\mathbb{R}$, with $I$ infinite. Convince yourself that $\operatorname{Hom}(M,\bigoplus_i N_i)$ corresponds to infinite matrices whose columns each have finitely many nonzero entries, while $\bigoplus_i \operatorname{Hom}(M,N_i)$ corresponds to infinite matrices with only a finite number of nonzero rows.

It's not quite clear to me why "$\bigoplus_i \operatorname{Hom}(M,N_i)$ corresponds to infinite matrices with only a finite number of nonzero rows".
I don't know how to write a formal rigorous proof that $\operatorname{Hom}_R\left ( M ,\bigoplus_{i\in I} N_{i}\right )\not \cong\bigoplus_{i\in I} \operatorname{Hom}_R\left ( M ,N_{i}\right )$ in this case.
 A: I think you're right that the vector spaces in this example will be isomorphic when $I$ is countably infinite. Probably what Mike Haskel had in mind is that the canonical map from the sum-of-Homs to the Hom-into-sum is not an isomorphism.  
The example becomes correct if $I$ is chosen more carefully.  What one needs is $|I|=\kappa$ for some cardinal $\kappa>|\mathbb R|$ with the additional property that $\kappa^{\dim M}>\kappa$.  For simplicity, let me take $\dim M=\aleph_0$ and all $N_i=\mathbb R$ for this example. The required cardinals $\kappa$ exist; indeed the inequality $\kappa^{\aleph_0}>\kappa$ holds whenever the cofinality of $\kappa$ is $\aleph_0$.
Having chosen $\kappa$ this way, we get for the dimension of the Hom-into-sum, $(|\mathbb R|\cdot\kappa)^{\aleph_0}=\kappa^{\aleph_0}$, while the sum-of-Homs has dimension $\kappa\cdot(|\mathbb R|^{\aleph_0})=\kappa\cdot|\mathbb R|=\kappa$. So the dimensions are different.
A: I think they are probably isomorphic to each other. Let me know where I am wrong.
The dimension of LHS is $|\bigoplus_{i=1}^{\infty}\mathbb R|^{\dim(M)}=|\mathbb R|^{\dim(M)}$, while the dimension of the RHS is $|\mathbb N||\dim(M^*)|=|\mathbb N||\mathbb R ^{\dim(M)}|=|\mathbb R ^{\dim(M)}|$.
Reference:
https://mathoverflow.net/questions/168596/dim-homv-w
