An ordinary category \(C\) is equivalent to a globally small category if and only if the family fibration \(\boldsymbol {\mathcal {F}}_{C}\) has a generic object.
Proof.
To see that this is the case, suppose that \(C\) has a set of objects. Then \(C\in \mathbf {Set}\) and we define \(\lfloor {C}\rfloor \) to be the displayed object \({\mathopen {}\left \{x\right \}\mathclose {}}_{x\in C}\in \boldsymbol {\mathcal {F}}_{C}[C]\). Fixing \(I\in \mathbf {Set}\) and \(z\in C^I\), we consider the cartesian map displayed over \(z : I \to C\):
Conversely assume that \(\boldsymbol {\mathcal {F}}_{C}\) has a generic object \(\bar {u}\in \boldsymbol {\mathcal {F}}_{C}[U]\) for some \(U\in \mathbf {Set}\); then we may equip \(U\) with the structure of a globally small category such that \(U\) is equivalent to \(C\), using the canonical cleaving of the family fibration. In particular, given \({1}\xrightarrow {{x,y}}{U}\) we define a morphism from \(x\) to \(y\) to be given by a vertical map \({x^{*}{\bar {u}}}\xrightarrow {{h}}{y^{*}{\bar {u}}}\) in \(\boldsymbol {\mathcal {F}}_{C}{\mathopen {}\left [1\right ]\mathclose {}}\simeq C\).
