Let be a functor such that the category has a terminal object . Then can canonically be factored as the composite of applied to the slice category , followed by dependent sum (projection on the source). We say that is a parametric right adjoint, or p.r.a., if the functor is a right adjoint. Parametric right adjoints are also called local right adjoints, though this terminology conflicts with that of local adjunctions (see locally). It is equivalent to the notion of right multi-adjoint, but multi-adjoints can be formulated without assuming has a terminal object. A monad is called p.r.a. if its functor part is p.r.a. and moreover its unit and multiplication are cartesian. Thus in particular it is a cartesian monad. A p.r.a. monad is also called a strongly cartesian monad. Since creates connected limits, if is p.r.a. then it preserves connected limits, and in particular preserves pullbacks. It follows that any p.r.a. monad is a cartesian monad. Conversely, a functor between presheaf categories is p.r.a. if it preserves connected limits. The reason is that (in the notation above), if preserves connected limits, then preserves all small limits, and a limit-preserving functor from a cototal category such as a presheaf category to a locally small category is a right adjoint. Any polynomial functor is p.r.a., since then can be identified with , which has the left adjoint . If is a presheaf category and is p.r.a., then the comma category (also called the Artin gluing in this context) is again a presheaf category. Conversely, if is a presheaf category, then preserves connected limits, and thus is p.r.a. A parametric right adjoint functor (with locally small codomain) has in particular a left multi-adjoint, which sends each object to the family of all units , where ranges over all morphisms and is the left adjoint of . This is because any morphism induces a unique composite , and hence a unique factorization through . Conversely, if has a terminal object and has a left...