I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states:
$\mathbf{Lemma A1.1.6}$
Let $M$ and $N$ be left $\Gamma$-comodules with $M$ projective over $A$ (where ($A$,$\Gamma$) is a Hopf algebroid). Then Hom$_A(M,A)$ is a right $\Gamma$-comodule.
In the proof it's stated that because $M$ is projective, there's a canonical isomorphism $$ Hom_A(M,A) \otimes_A N \approx Hom_A(M,N). $$
If I recall correctly, however, this is only true if $M$ is a finitely generated projective module, in which case you can localize at all primes and reduce it to the case of a finite rank free module.
Is there an implicit finite generation hypothesis on $\Gamma$-comodules that I'm missing, or is there some other way to define the comodule structure?