  We translate properties of the Sigma-type in Martin-L\"of Type Theory (MLTT)
to properties of the Grothendieck construction in category theory. Namely,
equivalences in MLTT that involve the Sigma-type motivate isomorphisms between
corresponding categories that involve the Grothendieck construction. The
type-theoretic axiom of choice and the "associativity" of the Sigma-type are
the main examples of this phenomenon that are treated here.
