Proposition. Discrete opfibrations as \(T\)-categories [mde-QIGV]

When \(T = (- + 1)\) is the maybe monad, the category of \(T\)-categories is equivalent to the category of discrete opfibrations of small categories and commutative squares.

Proof.  [#245]

A \((- + 1)\)-category \(O+1 \leftarrow M \rightarrow O\) amounts to a category \(C\) equipped with nullary morphisms, corresponding to \(O + 1 \leftarrow M\) landing in \(1\). This data can be arranged as a functor \(C \to \mathbf {Set}\), assigning to each object its set of nullary morphisms and to each morphism precomposition with it. In other words, we have a discrete opfibration over \(C\).