Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics – where it allows for synthetic (higher) category theory – and programming languages – where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathrm{𝒮︀}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathrm{𝒮︀}$ is directed univalent. The construction of $\mathrm{𝒮︀}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathrm{𝒮︀}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.