Corollary 3

Let $A$ be a ring, $(E_\alpha, f_{\beta\alpha})$ a direct system of right $A$*-modules and* $F$ a left $A$*-module. Then the $\mathbf{Z}$*-modules* $\lim (E_\alpha \otimes_A F)$ and $(\lim E_\alpha) \otimes_A F$ are canonically isomorphic.

It suffices to take $A_\alpha = A$ and $F_\alpha = F$ for all $\alpha \in I$ in Proposition 7.

In particular, if $\rho : A \to B$ is a ring homomorphism, $\lim \rho^*(E_\alpha)$ and $\rho^*(\lim E_\alpha)$ are canonically isomorphic.