### 3.3. Iteration and pushforward [000D]

It also makes sense to speak of categories displayed over other displayed categories; one way to formalize this notion is as follows. Let $E$ be displayed over $B$; we define a category displayed over $E$ to be simply a category displayed over the total category $\TotCat{E}$.

Now let $F$ be displayed over $E$ over $B$. Then we may regard $F$ as a displayed category $B\Sub{!}F$ over $B$ as follows:

1. An object of $(B\Sub{!}F)\Sub{x}$ is a pair $(\bar{x},{\ddot{x}})$ with $\bar{x}\in E\Sub{x}$ and ${\ddot{x}}\in F\Sub{\bar{x}}$.
2. A morphism $(\bar{x},{\ddot{x}})\to\Sub{f}(\bar{y},{\ddot{y}})$ is given by a pair $(\bar{f},{\ddot{f}})$ where $\bar{f}:\bar{x}\to\Sub{f}\bar{y}$ in $E$ and ${\ddot{f}}:{\ddot{x}}\to\Sub{\bar{f}} {\ddot{y}}$ in $F$.

By virtue of Section 3.2 [000B], we may define the pushforward of a displayed category along a functor. In particular, let $E$ be displayed over $B$ and let $U:B\to C$ be an ordinary functor; then we may obtain a displayed category $U\Sub{!}E$ over $C$ as follows:

1. First we construct the displayed category $U\Sub{\bullet}$ corresponding to the functor $U:B \to C$.
2. We recall that there is a canonical equivalence of categories $\TotCat{U\Sub{\bullet}}\to B$.
3. Because $E$ is displayed over $B$, we may regard it as displayed over the equivalent total category $\TotCat{U\Sub{\bullet}}$ by change of base (Section 2.8 ).
4. Hence we may define the pushforward $U\Sub{!}E$ to be the displayed category $(U\Sub{\bullet})\Sub{!}E$ as defined above.