I have found an interesting link between containers and the Grothendieck construction. Here's a sketch.

A container \(S \triangleleft P \) in a category \(\mathbb C\) consists of an object of shapes \(S\) and an arrow \(P \colon S \to \mathbb C\) of positions. For instance, in \(\mathbf{Set}\), the shape of lists is \(\mathbb N\) (lengths), and for each shape \(n \in \mathbb N \), positions are ordinals less than \(n\). To fill a shape with data, we construct the extension functor. E.g., in \(\mathbf{Set}\):
\[ T_{S \triangleleft P} X = \Sigma_{s \in S} (P s \to X) \]
(think of a list of \(X\)).
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