Application of universes: levels

In consistent proof assistants, we have for each level $i$, a universe ${\mathsf{Set}}_i:{\mathsf{Set}}_{i+1}$, ${\mathsf{El}}_i:{\mathsf{Set}}_i\to {\mathsf{Set}}_{i+1}$.

Typically universes are Russell-style which means that ${\mathsf{Set}}_i\subset {\mathsf{Set}}_{i+1}$ and ${\mathsf{El}}_i$ is the identity.

Type formers are natural with respect to ${\mathsf{El}}_i$.