coproduct, direct sum I thought about the problem of how to understand coproduct and direct sum and I think this could be thought of as a way of thinking. I am posting this to verify if my understanding is correct.
So what we need is the following.
Given data:
$$\begin{align}
&f_i\colon X_i\to P \\
&g_i\colon X_i \to X
\end{align}$$
Now define in someway
$$g\colon P \to X$$
$P$ is the coproduct we are trying to define. Please note that we need to define $g$ given $g_i$. $g$ is actually to be defined so that it completes the commutative diagram and then only $P$ is the coproduct.
The most obvious way of defining $g$ on each of the basis elements is $g := \sum g_i$ where $i$ ranges over the index set. 
Now all our group, ring homomorphism operations are defined over finite sums or finite products. So there is no way we could make sense of an infinite sum in the definition of $g$. Thus we have to restrict it to finite sum. Hence if we have to restrict it to finite sum then only a finite number of elements of $P$ could be non zero and the rest are all $0$s.
This is my understanding and if somebody could point out that there is some gaps it would be very helpful.
Thanks
 A: I saw you posted on MO, and it seems you are trying to relate the ideas of coproduct and direct sum.  Put simply, the notion of a coproduct is a generalization of the notion of a direct sum to a general category.  For concreteness, consider the case of vector spaces over a field $k$.  Given a $k$-vector space $W$ and two subspaces $U, V$, the sum $U+V$ is just $\{u+v|u\in{}U,v\in{}V\}$, and the sum is said to be direct if we have unique expression, i.e. if $u,u'\in{}U$, $v,v'\in{}V$ and $u+v=u'+v'$, then $u=u'$ and $v=v'$.  
Now, given a direct sum $U\oplus{}V$, there are 'injections' $i:U\to{}U\oplus{}V$ and $j:V\to{}U\oplus{}V$ taking elements to themselves, and we have the following universal mapping property:
Given a $k$-vector space $T$ and linear maps $f:U\to{}T$, $g:V\to{}T$, there is a unique linear map $[f,g]:U\oplus{}V\to{}T$ such that $[f,g]i=f$ and $[f,g]j=g$.
The map $[f,g]$ is defined by taking $u+v$ to $f(u)+g(v)$.  This is well-defined, since the sum is direct.  By the usual arguments, we can show that this universal property characterizes $U\oplus{}V$ up to isomorphism, so we have an abstract characterization of a direct sum purely in terms of maps.  Of course this characterization just says that $U\oplus{}V$, together with $i$ and $j$, is a coproduct diagram of $U$ and $V$ in $k$-Vect.
These remarks generalize to infinite direct sums as well, that is, given a family $(U_i)_{i\in{}I}$ of subspaces of $W$, the direct sum of these spaces, together with the relevant injection maps, is a coproduct diagram in $k$-Vect.
In a general category, it doesn't usually make sense to talk about elements and maps as we have above, but it always does make sense to talk about  morphisms, diagrams, and universal properties.  So, the notion of a coproduct really generalizes direct sums to the general categorical setting.
I hope this was helpful.
A: Perhaps this might work: you understand that in an abelian category the coproduct and the product of two objects are the same (when they both exist). You want to "understand" something about the coproduct and the product of infinite families and how they may differ.
In an abelian category, if $J\subseteq K$ are finite, then you have maps from $\prod_{j\in J}X_j$ to $\prod_{k\in K}X_k$ and back; the map from the $J$-indexed product to the $K$-indexed product is obtained by "extending by $0$s", while the map back is obtained by dropping the components not in $J$. These are in fact maps induced by the universal properties of the product and the coproduct by the canonical projections and immersions, recalling that the finite products are equal to the finite coproducts.
That means that given an infinite family $\{X_i\}_{i\in I}$, you can look at the family of finite products $\bigl\{\prod_{j\in J}X_j\mid J\subseteq I,\ |J|\lt\infty\bigr\}$ as both a directed family (ordered by the partial order of inclusion among finite subsets) and as an inversely directed family. If you take the direct limit of this family and interpret the objects as coproducts, then you get the coproduct of the full infinite family. If you take the inverse limit of this family and interpret the objects as products, you get the product of the full infinite family. 
The direct limit of the objects is defined by "local" conditions, so that intuitively you only need to worry about what is happening "locally", hence the objects should only require finitely many information; whereas the inverse limit requires a whole family of consistent objects, so that you need "a lot more" information. Intuitively, local information just isn't enough. 
For abelian groups, modules, and vector spaces, the directed limit consists of equivalence classes of objects of the form $[(x,J)]$ with $x\in \prod_{j\in J}X_j$, with $[(x,J)]=[(y,K)]$ if and only if there exists $M$ finite, $J,K\subseteq M$, and such that the images of $x$ and $y$ in $\prod_{m\in M}X_m$ agree; so elements are completely determined by information on a finite subset of $I$. 
On the other hand, the inverse limit is constructed at the set level as a subset of the set-theoretic product $\times(\prod X_i)$ of the families, taking those elements $(x_J)$ indexed by the finite subsets of $I$, such that if $J\subseteq K$ then the image of $x_K$ in $\prod_{j\in J}X_j$ is precisely $x_J$ ("consistent families"). This means that you may need information about what is happening on all coordinates to determine an element, and there are elements that are  not obtained merely by local information. 
(A possible model might be the difference between the directed limit of the cyclic groups $\mathbb{Z}_{p^n}$ and the inverse limit of the same groups; the directed limit is the Prüfer group, in which each element is, in a sense, "finite", whereas the inverse limit is the $p$-adic integers, in which you have elements that are not "finite", but rather "come" from 'infinite tuples'.)
