Bartosz Milewski on Nostr: I'm struggling with the definition of the category of elements--the direction of ...

I'm struggling with the definition of the category of elements--the direction of morphisms. Grothendieck worked with presheaves \(C^{op} \to \mathbf{Set}\), with a morphism \(a, x) \to (b, y)\) being an an arrow \(a \to b\) in \(C\). The question is, what is it for co-presheaves? Is it \(b \to a\)? nLab defines it as \(a \to b\) and doesn't talk about presheaves. Emily Riehl defines both as \(a \to b\), which makes one wonder what it is for \(C^{op}^{op} \to \mathbf{Set}\), not to mention \(C^{op}\times C \to \mathbf{Set}\).