  Two novel descriptions of weak {\omega}-categories have been recently
proposed, using type-theoretic ideas. The first one is the dependent type
theory CaTT whose models are {\omega}-categories. The second is a recursive
description of a category of computads together with an adjunction to globular
sets, such that the algebras for the induced monad are again
{\omega}-categories. We compare the two descriptions by showing that there
exits a fully faithful morphism of categories with families from the syntactic
category of CaTT to the opposite of the category of computads, which gives an
equivalence on the subcategory of finite computads. We derive a more direct
connection between the category of models of CaTT and the category of algebras
for the monad on globular sets, induced by the adjunction with computads.
