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 $\R f_*$ and if $f$ is etale, then $f^!$ is the inverse image $f^*$. We also show that $(-)^!$ is compatible with flat base change.