Transpose of Linear Transformation is a Linear Transformation

Theorem
Let $R$ be a commutative ring.

Let $G$ and $H$ be $R$-modules.

Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively.

Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Let $u \in \map {\LL_R} {G, H}$.

Let $u^\intercal: H^* \to G^*$ be the transpose of $u$.

Then $u^\intercal: H^* \to G^*$ is itself a linear transformation.

Proof
By definition of evaluation linear transformation:
 * $\forall x \in G: y \in H^*: \innerprod x {\map {u^\intercal} y} = \innerprod {\map u x} y$

Since we have:

and:

it follows that $u^\intercal: H^* \to G^*$ is a linear transformation.