Suppose $I,J$ represents any finite subsets of the set of natural numbers $N$ and ($∏_{i∈I}A_i,φ_{JI})_{i\subset j\subset N}$ be a directed set. Suppose, also that the the direct limit $(A,\varphi_I)_{I\subset N}$ of the system exists. My question is:
Let $\varphi:∏_{i∈N}A_i \to A$ and $\lambda:∏_{i∈N}A_i \to Y$ be maps and Suppose, $\varphi_I:∏_{i∈I}A_i \to A$, and $\lambda_I:∏_{i∈I}A_i \to Y$ be the restrictions of $\varphi$ and $\lambda$, respectively. Then, by the definition of direct limit, there exists a unique map $\gamma:A\to Y$ s.t $\gamma \circ \varphi_ I =\lambda_I$ for every finite subset of $N$ (Please see fig 1)
Does it follow from this that $\gamma \circ \varphi=\lambda$? (Fig 2).
Are there any deductions for the commutativity of maps from the commutative diagrams of directed system and direct limits, in the literature ?
With transition and restriction maps defined as you did in the comments then it is not the case that $\gamma\circ\phi = \lambda$ necessarily holds.
For example, in the category of sets the colimit $A$ is equal to the subset of $\prod_{i \in \mathbb N} A_i$ consisting of those $(a_i)_{i \in \mathbb N}$ for which $a_i = e_i$ holds for all but finitely many $i$. To define a map $\phi\colon\prod_{i \in \mathbb N}A_i \to A$ whose restriction to any $\prod_{i \in I}A_i$ is $\phi_I$ we simply define $\phi$ to be the identity on the subset $A$ and for those tuples not in the subset we can define their image however we please. In particular, this means there are many maps $\phi\colon\prod_{i \in \mathbb N}A_i \to A$ with the property that the restrictions agree with the $\phi_I$.
Now let $\phi, \lambda\colon\prod_{i \in \mathbb N}A_i \to A$ be two different maps whose restriction to any $\prod_{i \in I}A_i$ agrees with $\phi_I$. The map $\gamma\colon A \to A$ we get from the universal property is simply the identity map, so the equation $\gamma\circ\phi = \lambda$ reduces to $\phi = \lambda$, but this false! We explicitly chose $\phi$ and $\lambda$ to be different.