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 (𝐶ᵒᵖ)ᵒᵖ→𝐒𝐞𝐭 , not to mention (C^{op}\times C \to \mathbf{Set}).
