### 2.2. The generalized pullback lemma [0014]

In light of [0003], the following result for displayed categories generalizes the ordinary “pullback lemma.”

[001H] Lemma 2.2·a (Generalized pullback lemma).

Let $\bar{f} : \bar{x}\to\Sub{f}\bar{y}$, and suppose that $\bar{g} : \bar{y}\to\Sub{g}\bar{z}$ is cartesian over $g$. Then $\bar{f};\bar{g}$ is cartesian over $f;g$ if and only if $\bar{f}$ is cartesian over $f$.

Proof. Suppose first that $\bar{f}$ is cartesian. To see that $\bar{f};\bar{g}$ is cartesian, we must construct a unique factorization as follows: Because $\bar{g}$ is cartesian, we can factor $\bar{h} = i;\bar{g}$ for a unique $i:\bar{u}\to\Sub{m;f}\bar{y}$. Then, because $\bar{f}$ is cartesian, we can further factor $i = j;\bar{f}$ for a unique $j:\bar{u}\to\Sub{m}\bar{x}$. We conclude that there is a unique $j:\bar{u}\to\Sub{m}\bar{x}$ for which $\bar{h} = j;\bar{f};\bar{g}$, as required.

Conversely, suppose that $\bar{f};\bar{g}$ is cartesian. To see that $\bar{f}$ is cartesian, we must construct a unique factorization as follows: Because $\bar{f};\bar{g}$ is cartesian, we can factor $\bar{h};\bar{g} = i;\bar{f};\bar{g}$ for a unique $i:\bar{u}\to\Sub{m}\bar{x}$. On the other hand, because $\bar{g}$ is cartesian, there is a unique $j:\bar{u}\to\Sub{m;f}\bar{y}$ for which $\bar{h};\bar{g} = j;\bar{g}$; as both $\bar{h}$ and $i;\bar{f}$ satisfy this condition, we conclude $\bar{h}=i;\bar{f}$. Therefore, there is a unique $i:\bar{u}\to\Sub{m}\bar{x}$ for which $\bar{h} = i;\bar{f}$, as required.