Pasting pseudofunctors

We give an abstract criterion for pasting pseudofunctors on two subcategories of a category into a pseudofunctor on the whole category. As an application we extend the variance theory of the twisted inverse image (−)! over schemes to that over formal schemes. Thus we show that over composites of compactifiable maps of noetherian formal schemes, there is a pseudofunctor (−)! such that if f is a pseudoproper map, then f ! is the right adjoint to the derived direct image Rf∗ and if f is étale, then f ! is the inverse image f. We also show that (−)! is compatible with flat base change.