Given a diagram $D\colon\mathcal{C}\longrightarrow\mathcal{D}$, we say that a functor $J\colon\mathcal{I}\longrightarrow{C}$ is cofinal if we have a natural isomorphism $$ \mathrm{colim}\left(\mathcal{C}\overset{D}{\longrightarrow}\mathcal{D}\right)\cong\mathrm{colim}\left(\mathcal{I}\overset{J}{\longrightarrow}\mathcal{C}\overset{D}{\longrightarrow}\mathcal{C}\right). $$ With a number of ways to check cofinality for functors (e.g. 1, 2), this notion ends up being very useful in practice as a computational tool.
What about cofinality for natural transformations? Namely, given functors $D,D'\colon\mathcal{C}\rightrightarrows\mathcal{D}$, call a natural transformation $\eta\colon D\Rightarrow D'$ final if postcomposition with it induces a bijection¹ $$ \mathsf{Nat}\left(\Delta_{(-)},D\right)\overset{\eta}{\underset{\cong}{\dashrightarrow}}\mathsf{Nat}\left(\Delta_{(-)},D'\right), $$ and hence a natural isomorphism $\lim(D)\cong\lim(D')$. Do we have, as in the case of cofinality for functors, easy(-ish) to check conditions implying that a natural transformations is final?
[Motivation: If so, this would similarly be useful for either computations, or for expressing co/limits of somewhat contrived functors as co/limits of nicer, more naturally-appearing, ones. For example, having such a notion in hand, one would be able to e.g. compute the $\pi_0$ of a complicated simplicial set $S_\bullet$ by first finding a “final simplicial map” $S_\bullet\to T_\bullet$ to a simpler simplicial set $T_\bullet$ and then computing the $\pi_0$ of $T_\bullet$.]
¹Here's a picture of the induced cones of $D'$ over some object $X$ of $\mathcal{D}$:
