I am trying to make this question as clear as possible. I will have to elaborate a bit though in order to do so. 

I am in a first semester linear Algebra course (although people of a higher semster in my University claim that we're being taught Algebra rather than linear Algebra). This week I am struggling with the following problem:

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> **Current Problem Set** *(8th week)*: To show, for $(N,+,\cdot)$ being an R-Module $$\text{Hom}\_R(\oplus\_{i \in I}M\_i, N) \overset{\simeq}\to \Pi\_{i \in I} \text{Hom}\_R(M\_i,N) \\ \hom\_R(N, \Pi\_{i \in I} M\_i) \overset{\simeq}\to \Pi\_{i \in I} \hom\_R(N,M\_i) $$
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Which of course, as you can guess, I have no idea\* on how to approach 

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**To the Problem**: Exhausting google just a little bit, one does find many relevant links addressing the above problem, for example on this math blog (crazy project) [here](http://crazyproject.wordpress.com/2011/05/23/the-interaction-of-hom-with-direct-sums-and-direct-products/). However in the authors answer the person uses the universal property of the direct product. I have never been introduced to such a property in my course yet, which brings me to the word elementary that I have used in the title.

There are also related questions (I am not going to link them) on this website, the answers are of the caliber "trivial, indeed, it is easy to see that, it is clear that" by which I don't mean to mock the answerers - in fact they are in agreement to what my Professor would say to this exercise.

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**No idea\***: I fibbed there a little. On the level of definition I believe to know several things:

* The direct sum $\oplus\_{i \in I}M\_i$ is a sub-module of the direct product $\prod\_{i \in I}M\_i$ and the direct sum is defined to be $m\_i=0$ for almost all $ i \in I$ where $(m\_i)\_{i \in I}$ is a familiy of functions.
* The direct product is nothing but a set $M= \prod\_{i \in I} M\_i$ with a component wise defined addition and scalar multiplication for $(m\_i)\_{i \in I} \in M$
* An element $\varphi \in \hom\_R(\oplus\_{i \in I}M\_i,N)$ would be a homomorphism with scalars in $R$ that maps from $\oplus\_{i \in I}M\_i$ to $N$. So to make this a bit more precise: $$\varphi \in \hom\_R(\oplus\_{i \in I} M\_i, N) \implies \varphi : \begin{cases} \oplus\_{i \in I}M\_i & \longrightarrow N \\ (m\_i)\_{i \in I} & \longmapsto \varphi((m\_i)\_{i \in I}) \end{cases} $$
* An element of the 'direct product over the index set $I$ of all homomorphisms with scalars in $R$ from $M\_i$ to $N$' would be a 'family of mappings' of the like $$\omega\_i: \begin{cases}M\_i & \longrightarrow N \\ m\_i & \mapsto \omega\_i(m\_i) \end{cases} $$

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I am uncertain if all of the above is right. To quote my professor "all the exercises are trivial as soon as the definitions are clear". So I assume that I still lack understanding of the definition, because it feels hard for me to define an isomorphism as required in the exercise. 

If someone could explain to me the taught-process on how to obtain the isomorphism or link to me relevant literature where the matter is explained I would appreciate it. Please note that I want to work with the definitions as 'purely' as possible and not use many theorems (chances are high that I don't know them).