The following is taken from page 38 in A Term of Commutative Algebra by Altman and Kleiman. $\mathcal{C}$ and $\Lambda$ are categories, $\lambda\mapsto M_\lambda$ and $\lambda\mapsto N_\lambda$ are functors from $\Lambda$ to $\mathcal{C}$.
Assume $\mathcal{C}$ has direct limits indexed by $\Lambda$. Then, given a natural transformation from $\lambda \mapsto M_{\lambda}$ to $\lambda \mapsto N_{\lambda}$ , universality yields unique commutative diagrams
$$\require{AMScd} \begin{CD} M_\lambda @>>> \varinjlim M_\lambda\\ @VVV @VVV \\ N_\lambda @>>> \varinjlim N_\lambda \end{CD}$$
What exactly is this universality? And how does it yield unique commutative diagrams?
The universality here is the definition of direct limit.
Remember that the direct limit is an object $\varinjlim N$ together with a family of maps $\beta_\lambda : N_\lambda\to \varinjlim N$ (called "insertions" in your notes) that commutes with the "transition maps" of $N$; and a natural transformation $\gamma:M\to N$ is a family of maps $\gamma_\lambda:M_\lambda\to N_\lambda$ that commute with the "transition maps" of $M$ and $N$. So composing them gives you family of maps $\beta_\lambda\circ \gamma_\lambda :M_\lambda\to \varinjlim N$ that commutes with the transition maps. Then the universal property defining $\varinjlim M$ tells you that this family of maps must factor uniquely through the insertions $\alpha_\lambda:M_\lambda\to \varinjlim M$, or in other words, that there must be a unique map $\hat{\gamma}:\varinjlim M\to \varinjlim N$ (the right-hand vertical map in your diagram) such that $\hat{\gamma}\circ \alpha_\lambda=\beta_\lambda\circ \gamma_\lambda$ for all $\lambda$ (this is the commutativity of the diagram in your question).