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)?



Is it possible to pull back a natural transformation?
There is a good notion of a pullback transformation, but its domain and codomain aren't what you guessed; rather one asks for a map from $f^\*(g\_0) \to f^\*(g\_1)$ and a 2-cell filling the resulting triangle over G. The existence of such a pullback transformation is also not automatic from the existence of pullbacks, but requires $f$ to be a fibration (and the existence of all such pullbacks is equivalent to $f$ being a fibration). This characterization of fibrations is studied in Peter Johnstone's paper "Fibrations and partial products in a 2-category".


