Edit 2: Maybe if I explain the origin of my question, it will make clearer what I was asking. When trying to define a semantics of DTT in a locally cartesian closed category, it seems to me that the natural approach is to define the interpretation of each type and term inductively using the corresponding categorical constructions, as in D4.4 of Sketches of an Elephant. Unfortunately, this doesn't quite work, because for instance substitition is "strict" in type theory, but pullback is only up to isomorphism in a category.
It seems that the standard way to deal with this is to first strictify the pullbacks in the lccc, e.g. by replacing its codomain fibration by a split one and maybe some additional data (people talk about "comprehension categories" and suchlike). However, looking at the direct construction that doesn't work, it seems to me that it can be interpreted meaningfully as an inductive construction acting not on types+terms, but rather on derivations of typing judgments. If that's right, it seems like one could then get a meaningful interpretation of type theory in an lccc by proving that any two derivations of the same typing judgment produce canonically isomorphic interpretations in the lccc. But that would require a notion of equivalence between derivations which is tractable (so that one can show that any equivalence of derivations gives an isomorphism in the lccc) and under which all derivations can be shown to be equivalent; hence my question.
