Suppose that a 2-category $\mathcal{C}$ has strict pullbacks and one has maps $f:F\to C$, $g_0,g_1:G\to C$ and a natural transformation $\gamma:g_1\implies g_0$. Is there a good notion of a pullback transformation $f^*\gamma$. If so, I would expect its codomain to be $$f^*g_1\times_G f^*g_0\to f^*g_1\to F$$ and similarly for the domain.
If there is, is the need for the extra pullback here (over $G$) connected to the second layer of pullbacks in a 2-categorical descent diagram (relative to quotients in a 1-cat, which only require a kernel pair=1 layer of pullbacks)?