Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that $F \Box_{A//B} A$ is then an $A$-free coresolution of $k \Box_{A//B} A$. I would appreciate any help with the following questions. Why does cotensoring with $A$ over $A//B$ preserve exactness here? Is $- \Box_{A//B} A$ always an exact functor?
(I am attempting to follow Kochman's book in a computation of the homotopy of some Thom spectra, which is where the above comes up)