Diagrammatic presentation of instances of models

[epatters]

A diagram in a category $$C$$ presents an instance of (copresheaf on) $$C$$ in either of the following equivalent ways:

1. By post-composing the diagram with the Yoneda embedding and taking the colimit of representables (see e.g. Kan extensions are partial colimits, Section 3)
2. By using the comprehensive factorization to factor the diagram as an initial functor followed by a discrete opfibration, then taking the latter

Constructing the category of elements (cf. #239) of (1) gives (2). In fact, this is how the nLab establishes the comprehensive factorization.

Generalizing from the case of the discrete double theory, we define a diagram in a model of a discrete double theory to be any model morphism into that model. The task is then to show how a diagram in a model presents an instance of the model. This could be accomplished by generalizing either/both of (1) and (2).