By mimicking the internal sheafification construction in an arbitrary topos, we provide a simple description of the various sheaf-related constructions in a purely type-theoretical setting. Assuming mild extensionality properties on our target type theory, we show that it almost results in a syntactic model of CIC. Unfortunately, a well-known topos-theoretic issue carries over, and the resulting theory, called ShTT, does not feature universes. We propose three different solutions to this problem: making the target theory univalent, allowing effects in the source theory, or changing our point of view while using strict propositions. As a side-product, we also give a computational interpretation of sheaves that highlights their deep relationship with the well-known structure of interaction trees.