Direct limit of a directed system I am trying to compute direct limit of a directed system.
I understand the definition but I did not really compute what is a direct limit of a directed system given one such directed system.

Let $\mathcal{C}$ be a category and $I$ be a directed set.  A
  collection of objects $\{F_i\}$ indexed by elements of $I$ along with
  morphims $f_{ij}:F_i\rightarrow F_j$ indexed by elements $(i,j)$ with
  $i\leq j$ is said to be a directed system if 
  
  
*
  
*$f_{ii}$ is identity on $F_i$.
  
*For $i\leq j\leq k$ we have $f_{ik}=f_{jk}\circ f_{ij}$ in $F_i\xrightarrow{f_{ij}}F_j\xrightarrow{f_{jk}} F_k$. 
  
  
  Let $\mathcal{C}$ be a category and $(F_i,f_{ij})$ be a directed system in  $\mathcal{C}$. 
  An object $G$ in $\mathcal{C}$ along with morphisms $\phi_i:F_i\rightarrow G$ for each $i\in I$ is called a direct limit 
  if $\phi_i=\phi_j\circ f_{ij}$ for all $i,j$ with $i\leq j$ and, given any object $H$ along with morphisms $\psi:F_i\rightarrow H$ for each
  $i\in I$ with $\psi_i=\psi_j\circ f_{ij}$, there exists a unique morphism $\Phi:G\rightarrow H$ such that $\psi_i=\Phi\circ \phi_i$ for all 
  $i\in I$.

Let $A_1\xrightarrow{f_{12}}A_2\xrightarrow{f_{23}}A_3\rightarrow \cdots$ be a directed system indexed by natural numbers. Other maps $A_m\rightarrow A_n$ for $m\leq n$ is just the composition $A_m\rightarrow A_{m+1}\rightarrow \cdots\rightarrow A_n$.
Now, I need to construct a set $B$ with maps $A_i\rightarrow B$ for all $i\in \mathbb{N}$ with some universal property. 
I guess $B$ is the disjoint union of $A_i$. Say $B=\bigcup_{n\in \mathbb{N}}{n}\times A_n$. We have maps $A_i\xrightarrow{\phi_i} B$ by $x_i\mapsto (n,x_i)$. We need to have $\phi_i(x_i)=\phi_j(f_{ij}(x_i))$ for all $i\leq j$.
We need to have $(i,x_i)=(j,f_{ij}x_i)$ for all $i\leq j$. As this is always not true in $B$, we consider the quotient of $B$ by these conditions i.e., $(i,x_i)=(j,f_{ij}(x_i))$ for all $i\leq j$ and call this quotient $C$. So, 
$$C=\left(\bigsqcup_{\{n\}\in \mathbb{N}}{n}\times A_n\right){\bigg /}\sim$$
where $(i,x_i)\sim (j,f_{ij}(x_i))$.
I claim this is the direct limit of $(A_i)$. I could not prove the universal property. Please give hints for that and comment whether $C$ is a right choice or not.
 A: In the category of sets, $B$ is not exactly the disjoint union of the $A_i$s (which is the coproduct in $\operatorname{Set}$).
It is the quotient of $\coprod_\limits{i\in\mathbf N} A_i$ by the equivalence relation:
$$x_i\sim x_j\iff\exists k\ge i,\,;f_{ik}(x_i)=f_{jk}(x_j).$$
Note that, in case the $A_i$s are subsets of a larger set and the morphisms are the inclusion morphisms, the direct limit is nothing else than the union of all $A_i$s.  From this point of view, we can say direct limits are a vast generalisation of set-theoretic union, in the context of category theory.
A: Your $C$ is correct. Let $\pi\colon B\to C$ denote the quotient map.
Let $H$ be a set together with maps $\psi_i\colon A_i\to H$ such that $\psi_{i+1}\circ f_{i,i+1}=\psi_i$ for all $i$. 
Then we can define a map $\Phi\colon C\to H$ as follows:
Given $c\in C$, pick $b\in B$ with $\pi(b)=c$. Then $b=(n,a)$ for aome $n\in\Bbb N$ and $a\in A_n$. Set $\Phi(c):=\psi_n(a)$. This is well-defined: If also $\phi(b')=c$ for some $b'=(n',a')$ with $a'\in A_{n'}$, then $\psi_{n'}(a')=\psi_n(a)$ because (say, in the simplest case that $n'=n$) $a'=f_{n,n'}(a)$ and so $\psi_{n'}(a')=\psi_{n'}(f_{n,n'}(a))=\psi_n(a)$.
It is also clear that we must define $\Phi$ as described above, i.e., $\Phi$ is unique.
A: You have the correct object $C$. Here's a proof of the universal property of the colimit (direct limit) using universal property of disjoint sums and quotient.
First notice that disjoint sum in $\textbf{Set}$ is the coproduct. And taking quotient by a relation has the following universal property : given an equivalence relation $\sim$ on $X$, we have a morphism  $\pi:X\to X/_\sim$ such that for any morphism $f:X\to Y$, satisfying $a\sim b\implies f(a)=f(b)$, there exists a unique morphism $\tilde{f}:X/_\sim\to Y$, with $f = \tilde{f}\circ\pi$.
Now, you have defined $B=\coprod F_i$ and $C=B/_\sim$, where $\sim$ is the equivalence relation given by the directed system. To show that $C$ is the colimit, say you have a collection of morphism $\psi_i:F_i\to H$, which is compatible with the direct system. Firstly, from the coproduct, you get a unique morphism $\psi:B\to H$. Notice that $b\sim b'\implies\psi(b)=\psi(b')$. So again by the universal property of the quotient, you get a unique map $\Psi:C\to G$, as required. Thus, $C$ is the colimit.
