Now replace sets with categories. A categorical container is a category of shapes \(S\) and a (pseudo-) functor \(P \colon S \to \mathbf{Cat}\). It's the same functor that appears in the Grothendieck construction.
An arrow from \(S \triangleleft P \) to \(T \triangleleft Q \) is a functor \(S \to T\) and a natural transformation \(Q \to P\).
We can fill the container with data from a category \(X\) using a coend:
\[ T_{S \triangleleft P} X = \int^{s \colon S} (P s \to X) \]
2/3
An arrow from \(S \triangleleft P \) to \(T \triangleleft Q \) is a functor \(S \to T\) and a natural transformation \(Q \to P\).
We can fill the container with data from a category \(X\) using a coend:
\[ T_{S \triangleleft P} X = \int^{s \colon S} (P s \to X) \]
2/3
1