Vtyjkyuk

I'm in trouble with the definition of generator of $Rtext-mathbfmod$ category of left $Rtext-mathbfmod$. I've found a lot of equivalent definitions. I'm using the one of Jacobson's Basic Algebra, so $X$ is a generator if every module $M$ is sum of submodules all of which are homomorphic images of $X$.

Does it mean that these submodules are images of $X$ via homomorphisms?

Then I've proved that the following are equivalent:

1) $X$ is a generator;

2) the functor $operatornamehom(X, -)$ is faithful;

3) $T(X)=R$ where $T(X) = sum_h in operatornamehom(X,R) h(X)$;

4) $R$ is a homomorphic image of $X^n$ for some $n$.

Now I need to use the fact that $X$ is a generator if and only if every left $R$-module $M$ is an epimorphic image of some direct sum $bigoplus_i X$.
Is this fact only a different way to write the definition I've used? I don't know what an epimorphic image is and I'm confused about this sentence.

If anybody can help me this will be very appreciated! Thanks!

If $R$ is a unital ring, you know that every module $M$ can be described as a homomorphic image of some free module $F$; since
$$
F cong bigoplus_i in IR
$$
for some index set $I$, you are pretty much done. Explicitly, let $M$ be a left $R$-module and let $F$ be a free module chosen such that you can realize $M$ as the homomorphic image of some map $phi:F to M$, i.e., $operatornameImage(phi) cong M$. Then since there exists an $n in mathbbN$ such that $R$ is the homomorphic image of $X^n$, you get that
$$
F cong bigoplus_i in I R cong bigoplus_i in I X^n cong bigoplus_i in Ileft(bigoplus_k=1^nXright) cong bigoplus_j in J X
$$
where $I$ is some index set and $J$ is the evident new index set given by reindexing the double direct sum. Call the isomorphism from $F$ to this direct sum above $alpha$, i.e.,
$$
alpha:bigoplus_j in J X to F.
$$
Then pre-composing $phi$ with $alpha$ gives a surjection
$$
(phicirc alpha):bigoplus_j in J X to M
$$
where
$$
operatornameImage(phi circ alpha) = operatornameImage(phi) = M
$$
because $alpha$ is an isomorphism and $phi$ is a surjection.