Lemma 1

The mapping $c''_E$ is bijective. If E is finite-dimensional, then so is $c'_E$.

When F is a right A-module and G is a K-vector space, we defined in II, §4, No. 1, p. 268, Proposition 1, a) an isomorphism $\gamma$ of K-vector spaces from Hom$_K(F\otimes_AE,G)$ to Hom$_A(E$, Hom$_K(F,G))$. This isomorphism sends $\varphi : F\otimes_AE\rightarrow G$ to the homomorphism $e\mapsto (f\mapsto \varphi (f\otimes e))$. Through the canonical isomorphism from $A_d\otimes_AE$ to E (II, §3, No. 4, p. $249$)$,\gamma$ can be identified with $c''_E$ when $F = A_d$ and G = K.

Analogously, the isomorphism $\beta$ defined in II, §4, No. 1, p. 268, Proposition 1, b) specializes to an isomorphism $\beta$ from Hom$_K(E^*\otimes_AA_s,K)$ to Hom$_A(E^*$, Hom$_K(A_s,K))$. When E is finite-dimensional, E can be canonically identified with Hom$_K(E^*,K)$ and $E^*$ with $E^*\otimes_AA_s$; the homomorphism $\beta$ is then identified with $c'_E$.