I don't understand the following definition in Illusie, Complexe Cotangent et Déformations II (Springer LNM 283), page 17:
Soit $T$ une catégorie possédant des produits fibrés. On appelle objet en catégories (ou catégorie) dans $T$ tout préfaisceau $X$ sur $T$ à valeur dans (Cat) tel que ob($X$) et fl($X$) soient représentables.
I suppose that a presheaf $X$ on $T$ with values in (Cat) is simply a contravariant functor, but I do not know then what are the objets and morphisms of $X$, nor what it means that they are representable.
You are correct that "presheaf on $T$ with values in (Cat)" means precisely a contravariant functor $X:T\to \mathbf{Cat}$. Thus for every object $t$ of $T$, you have a category $X(t)$; this category has a set of objects $O(t)=ob(X(t))$ and a set of morphisms $M(t)=fl(X(t))$. Then $O$ and $M$ define contravariant functors $T\to \mathbf{Set}$, and this is what Illusie calls $ob(X)$ and $fl(X)$. To say that they are representable is simply to say that they are so as contravariant functors, i.e. there must be objects $t_o$ and $t_m$ of $T$ such that $O\cong \operatorname{Hom}_T(\_,t_o)$ and $M \cong \operatorname{Hom}_T (\_,t_m)$.
In particular, since the Yoneda embedding is full, there must be two arrows $c,d:t_m\to t_o$ and an arrow $i:t_o\to t_m$ that induce the codomain and domain natural transformations $fl(X)\to ob(X)$ and the "identity arrow" natural transformation $ob(X)\to fl(X)$; and since it also preserve and reflects limits, there must be an arrow $\mu:t_m\times_{t_o}t_m\to t_m$ that induce the "composition" natural transformation $fl(X)\times_{ob(X)}fl(X)\to fl(X)$. And since the Yoneda embedding is faithful, all these must satisfy certain conditions corresponding to the requirements that the composition is associative, unital and has the appropriate domains and codomains. This means that $(t_m,t_o,c,d,i,\mu)$ must define an internal category or "category object" in $T$.