I've been trying to gather definitions and results to understand the internal Diaconescu theorem, for I wish to read Joyal-Tierney's An Extension of the Galois Theory of Grothendieck. My question regards a passage in the Elephant which shows exactly what I was looking for, but I can't understand the implicit details. I'll give some context before.
One notion of internal presheaf which I was already acquainted with comes from Sheaves in Geometry and Logic, it is a map $\pi: F \to C_0$ equipped with a right action $\mu: F \times_{C_0} C_1 \to F$ satisfying some diagrams.
Before looking at the Elephant, I tried defining Yoneda as a functor (where the domain is the domain of the externalization of the internal category) $\underline{\mathbb{C}} \to \mathrm{Presh(\mathcal{\mathbb{C}})}$, by constructing a sort of slice category as the domain ($F = C_1 \times_{C_0} I$), and the definition seemed pretty solid, except for the fact I couldn't see the codomain (the category of internal presheaves) as a fibration.
Then I looked at the Elephant (part B) and there I found the definition of internal diagram (Definition B.2.3.11), s.t. if we take $\mathcal{S}$ to be indexed over itself, $\mathbb{C}$-shaped diagrams in $\mathcal{S}$ (whose category is denoted by $\mathcal{S}^\mathbb{C}$) correspond to the notion of presheaves I mentioned. Now, given an indexed category $\mathcal{D}$, we have an indexed category $\mathcal{D}^\mathbb{C}$ with $\mathcal{D}^\mathbb{C}(I) = \mathcal{D}^{\mathbb{C} \times I}$ where $I$ is taken as a discrete category.
This seemed thus the appropriate codomain for the Yoneda functor, which is presented in the Elephant by first going through discrete opfibrations, which were internal functors $F: \mathbb{F} \to \mathbb{C}$ s.t. the square with $F_0, F_1$ was a pullback. Then in the Elephant $x: I \to C_0$ gives rise to a simplicial object $C_n \times_{C_0} I$ and thus a category $\mathbb{R}{(x)}$. The projections $C_n \times I \to C_n$ give rise to an opfibration. I was following until it is stated that since all structure maps (I guess these are the face and degeneracy maps) are over $I$, we have that the resulting category $\mathbb{R}(x)$ is a $\mathbb{C}$-shaped diagram in $\mathcal{S}/I$, that is, belongs to $\mathcal{S}^\mathbb{C}(I) = \mathcal{S}^{\mathbb{C} \times I}$. I can't understand what the author means here. This is in Example B.2.5.4 (c).
What exactly is the $\mathbb{C}$-shaped diagram in $\mathcal{S}/I$? I can't make sense of this. It seems the book is referring to the category $\mathbb{R}(x)$ itself, but in what sense is this object a diagram/presheaf? I say presheaf since I'm trying to look at the internal presheaves = discrete opfibrations over $\mathbb{C}$ = $\mathbb{C}$-shaped diagrams in $\mathcal{S}$ correspondence to see if by $\mathbb{R}(x)$ the book refers to the associated presheaf/fibration, but then if it is the case how is it over I?
I suspect I might be missing something obvious since I'm very new to all these definitions, but I really wanna understand how to construct the Yoneda functor as an indexed/cartesian functor. Any help is greatly appreciated. I know this is somewhat vague but I'll provide any further definition/reference if needed.
EDIT: I think I figured out what was meant there, but I'd appreciate if someone could check my reasoning. I actually thought of this possibility before but not so much as to consider it seriously.
Recall that a $\mathbb{C}$-shaped diagrams over $\mathcal{S}$ is $\pi: F \to C_0$ with $\Phi: F \times_{d_0} C_1 \to F \times_{d_1} C_1$. From there we can recover the action $\mu: F \times_{d_0} C_1 \to F$. Conversely, the action yields $(\mu, p_1): F \times_{d_0} C_1 \to F \times_{d_1} C_1$
Now, in our particular case, we have $\pi: C_1 \times_{d_1} I \to C_0$ with action $\mu: C_1 \times C_1 \times I \to C_1 \times I$. Recalling that $(\mathcal{S}/I)^X \cong \mathcal{S}/(I \times X)$, our diagram is comprised of $\pi = 1 \times d_1: I \times C_1 \to I \times C_0$ and $\Phi$ is given by
Equivalently, speaking of (co)-presheaves as actions, we consider $\pi: I \times C_1 \to C_0$ with its action as being over the product $\mathbb{C} \times I$ instead with $1_I \times \pi$. Is this how the Yoneda functor was defined in the book?
Also, just to clarify (I know I'm asking a second question, but it kinda fits here), the indexed category definition for $\mathcal{S}^\mathbb{C}$ is induced by the geometric morphism $\mathrm{Copresh}{(\mathbb{C})} \to \mathcal{S}/C_0 \to \mathcal{S}$, right?
Not really, though it is induced by the functor taking $I$ to the discrete path-action $(I \times C_0 \to C_0, 1 \times d_1: I \times C_1 \to I \times C_0)$.