category object in an (∞,1)-category, groupoid object
Given a finitely complete category $E$, one can consider the bicategory $Cat(E)$ of internal categories in $E$, and thus internal functors, which are the morphisms in $Cat(E)$. If $E = Set$ then one can consider not only functors among small categories but also functors of the type $F: C\to Set$ from a small category $C$ to a large category of sets. In that case one can describe $F$ as consisting of a $C_0$-indexed family of objects and an action of $C_1$ on the diagram.
Compare the ideas discussed on this page with those at internal profunctor and discrete fibration. All three notions intersect — an internal diagram on $C$ is the same thing as an internal profunctor $C ⇸ 1$, which is the same thing as a discrete opfibration in $Cat(E)$. The three generalize the basic idea in different ways.
Given an internal category $C\in Cat(E)$, with the usual structure maps $s,t,i,c$, an internal diagram $F$ on $C$ (or, of type $C$) is given by
where $F_1$ is the pullback
These data must satisfy the following conditions:
$e$ respects the source and target maps of $C$, in that $d \circ e = t \circ s^* d$. Equivalently, $e$ is a morphism from $s^* d$ to $t^* d$ in $E/C_1$.
$e$ is an action in the sense that $e \circ (1 \times i) = 1$ and $e(e \times 1) = e(1 \times c)$.
It is clear how to define homomorphisms of internal diagrams: a morphism $F \to G$ is given by an $E/C_0$-morphism $F_0 \to G_0$ that commutes with the actions $e$. Internal diagrams on $C$ in $E$ form a category denoted by $E^C$.
An internal diagram on $C^{op}$ is sometimes called an internal presheaf on $C$.
Using the language of dependent types, the map $d: F_0 \to C_0$ can be seen as the interpretation of a dependent type $(X:C_0) \,\vdash\, (F(X):Type)$. The action of $C_1$ on $F_0$ can equivalently be given by the interpretation of a term in context:
Here we consider $C_1$ to depend on $C_0 \times C_0$ by the canonical morphism $C_1 \to C_0 \times C_0$ induced by $s$ and $t$, as in the type-theoretic definition of category.
If the ambient category $E$ is a locally cartesian closed category, so that its internal type theory has dependent product types, then this can be interpreted instead as a closed term
The axioms then take a particularly familiar form, also to be interpreted in the internal language of $E$:
From an internal diagram $(F,C,\lambda,e)$ one can equip $F =(F_0,F_1)$ with a structure of an internal category over $C$. In other words, there is a forgetful functor $E^C\to Cat(E)/C$ (where $Cat(E)/C$ is the corresponding slice category). This functor is fully faithful and its essential image consists precisely of all objects in $Cat(E)/C$ which are discrete opfibrations. Similarly, the objects of $E^{C^{op}}$ are the discrete fibrations in $Cat(E)/C$.
There is a composite forgetful functor $U \colon E^C \to Cat(E)/C \to E/C_0$. This functor $U$ is monadic — its left adjoint takes $p \colon X \to C_0$ to $t \circ s^* p \colon X \times_{C_0} C_1 \to C_1 \to C_0$.
An internal diagram as above may take values in any Grothendieck fibration over $E$. Given a fibration in the guise of an indexed category $F : S^{op} \to Cat$, a $C$-diagram in $F$ is given by
satisfying ‘cocycle equations’
modulo coherent isos, where the $p_i$ are the projections out of $C_2$.
By the Yoneda lemma for bicategories, the object $P$ determines (up to canonical isomorphism) a pseudonatural $\alpha^0 : E(-,C_0) \to F_0$ in $[E^{op},Cat]$, where $E$ is considered as a locally discrete bicategory, and $F_0(X) = ob F X$ considered as a discrete category, such that $\alpha^0(f) \cong f^* P$. Similarly, $\phi$ determines $\alpha^1 : E(-,C_1) \to F_1 = arr \circ F$, and $\alpha^1(g) \cong g^* \phi$. It is not hard to check that the conditions above correspond to requiring that the $\alpha^i_X$ form a functor $E(X,C) \to F X$ for each $X$, and pseudonaturality then makes the $C$-diagram $(P,\phi)$ equivalent to an indexed functor $E(-,C) \to F$. The category of $C$-diagrams in $F$ is then simply the hom-category $[E^{op},Cat](E(-,C),F)$.
An internal diagram on $C$ in the sense above is a $C$-diagram in the codomain fibration of $E$, that is the pseudofunctor $X \mapsto E/X$.
If $E$ is equipped with a coverage and $C$ is the Cech nerve associated to a cover $p : U \to X$ in $E$, then the category of $C$-diagrams in $F$ is the descent category $Des_p(F)$.
P. T. Johnstone, Topos theory, 1977, chapter 2
Elephant, B2.3