I'm not sure exactly what you're asking, but the general pattern for "positive" types is to assert that every "structured" display map over the type has a specified structured section.  This matches the induction principle of a positive type former in type theory.  One then needs to require a sort of "pullback-stability" condition as well in order to model substitution in type theory.  A forthcoming paper of Lumsdaine and Warren will describe a general process of "strictification" of "weakly stable" structure in a comprehension category into "strictly coherent" structure in a fully strict structure.

For identity types, the categorical structure is essentially a well-behaved weak factorization system in which the display maps are (or generate) the right class: see for instance [this paper](http://homotopytypetheory.org/references/awodey-warren-homotopy-theoretic-models-of-identity-types/).