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Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, ยง7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and morphisms $d_2^0, d_2^1, d_2^2 : C_2 \to C_1$, $d_1^0, d_1^1 : C_1 \to C_0$, $s_0^0 : C_0 \to C_1$, satisfying the following fragment of the simplicial identities: \begin{align} d_1^0 \circ d_2^0 & = d_1^0 \circ d_1^1 \\ d_1^0 \circ d_2^2 & = d_1^1 \circ d_2^0 \\ d_1^1 \circ d_2^1 & = d_1^1 \circ d_2^2 \\ d_1^0 \circ s_0^0 & = \textrm{id}_{C_0} \\ d_1^1 \circ s_0^0 & = \textrm{id}_{C_0} \end{align} If you squint hard enough, you will see that any internal category in $\mathcal{S}$ gives rise to a precategory: \begin{align} d_1^0 & = \text{codomain} \\ d_1^1 & = \text{domain} \\ s_0^0 & = \text{identity arrow} \\ d_2^0 & = \text{second arrow of a composable pair of arrows} \\ d_2^1 & = \text{composite of a composable pair of arrows} \\ d_2^2 & = \text{first arrow of a composable pair of arrows} \end{align}

Now, Borceux and Janelidze go on to define the category of "covariant presheaves" (i.e. a copresheaf) on $\mathbb{C}$ as the pseudolimit (or bilimit... I'm not entirely certain) of the induced pseudocommutative diagram of pullback functors

$$(\mathcal{S} \downarrow C_2) \mathrel{\hbox{$\begin{matrix} \smash{\leftarrow} \newline \smash{\leftarrow} \newline \smash{\leftarrow} \end{matrix}$}} (\mathcal{S} \downarrow C_1) \mathrel{\hbox{$\begin{matrix} \smash{\leftarrow} \newline \smash{\rightarrow} \newline \smash{\leftarrow} \end{matrix}$}} (\mathcal{S} \downarrow C_0)$$ but can we describe this category explicitly? It seems to me that a copresheaf on $\mathbb{C}$ is determined by a morphism $p_0 : F_0 \to C_0$ in $\mathcal{S}$ together with a choice of pullbacks $$\begin{array}{rcl} F_1 & \overset{\hat{d}_1^0}{\to} & F_0 \\ {\scriptstyle p_1} \downarrow & & \downarrow {\scriptstyle p_0} \\ C_1 & \underset{d_1^0}{\to} & C_0 \end{array} \hspace{3.0ex} \begin{array}{rcl} F_0 & \overset{\hat{s}_0^0}{\to} & F_1 \\ {\scriptstyle p_1} \downarrow & & \downarrow {\scriptstyle p_0} \\ C_0 & \underset{s_0^0}{\to} & C_1 \end{array} \hspace{3.0ex} \begin{array}{rcl} F_1 & \overset{\hat{d}_1^1}{\to} & F_0 \\ {\scriptstyle p_1} \downarrow & & \downarrow {\scriptstyle p_0} \\ C_1 & \underset{d_1^1}{\to} & C_0 \end{array}$$ $$\begin{array}{rcl} F_2 & \overset{\hat{d}_2^0}{\to} & F_1 \\ {\scriptstyle p_2} \downarrow & & \downarrow {\scriptstyle p_1} \\ C_2 & \underset{d_2^0}{\to} & C_1 \end{array} \hspace{3.0ex} \begin{array}{rcl} F_2 & \overset{\hat{d}_2^1}{\to} & F_1 \\ {\scriptstyle p_2} \downarrow & & \downarrow {\scriptstyle p_1} \\ C_2 & \underset{d_2^1}{\to} & C_1 \end{array} \hspace{3.0ex} \begin{array}{rcl} F_2 & \overset{\hat{d}_2^2}{\to} & F_1 \\ {\scriptstyle p_2} \downarrow & & \downarrow {\scriptstyle p_1} \\ C_2 & \underset{d_2^2}{\to} & C_1 \end{array}$$ such that the "hatted" version of the simplicial identities are satisfied. But this seems to be saying that a copresheaf on a precategory is another precategory, and that doesn't sound right at all.

So, what is a copresheaf on a precategory?

Edit. After some further thought, I now see that if $\mathbb{C}$ is actually an internal category in $\mathcal{S}$, then the above defines a copresheaf on which $\mathbb{C}$ acts invertibly, and the internal category structure apparent in this definition is exactly the category of elements of the copresheaf. (In other words, we are identifying copresheaves with discrete opfibrations.) So maybe this is correct after all...

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