Suppose $\mathcal{F}: \mathscr{C} \rightarrow \mathscr{D}$, $\mathcal{G}: \mathscr{E} \rightarrow \mathscr{D}$, $\mathcal{G}': \mathscr{E} \rightarrow \mathscr{D}$ are functors such that $\mathcal{G}$ and $\mathcal{G}'$ map into the essential image of $\mathcal{F}$. Suppose, furthermore, that $\mathcal{G}$ and $\mathcal{G}'$ pull back (in the topologist's sense) to functors $\mathcal{H}: \mathscr{E} \rightarrow \mathscr{C}$, $\mathcal{H}': \mathscr{E} \rightarrow \mathscr{C}$, i.e. $\mathcal{G} = \mathcal{F} \circ \mathcal{H}$, $\mathcal{G}' = \mathcal{F} \circ \mathcal{H}'$.
Let $\alpha: \mathcal{G} \rightarrow \mathcal{G}'$ be a natural transformation. If $\mathcal{F}$ is fully faithful, then it is possible to also pull back $\alpha$ to a natural transformation $\beta: \mathcal{H} \rightarrow \mathcal{H}'$ such that $\text{id}_\mathcal{F} \circ_h \beta = \alpha$ where $\circ_h$ denotes the horizontal composition of functors, as $\mathcal{F}$ being fully faithful guarantees that all component maps of $\alpha$ are also in the essential image of $\mathcal{F}$.
On the other hand, if $\mathcal{F}$ is not fully faithful, one cannot guarantee that all morphisms involved in $\alpha$ are in the essential image of $\mathcal{F}$ even when knowing that all of $\mathcal{G}$ and $\mathcal{G}'$ are.
Given this set up, is there a term for whether or not $\alpha$ can be pulled back, or is there literature on conditions that necessitate that $\alpha$ can be pulled back (outside of something abundantly nice like $\mathcal{F}$ being fully faithful)?