Definition

Let $R$ be the ring. The Hom on $k G$ -module category is the map.

Ho m_{R} (-, -) : (k G - m o d)^{o p} \times (k G - m o d) \to k G - m o d .

Different in Base

Let $G$ be a finite group and $k$ be a field. Let $V$ and $W$ be $k G$ -modules. If $R = k G$ , then $Ho m_{k G} (V, W) = {f : V \to W ∣ f is a k G -module homomorphism}$ is a $k$ -vector space.
Same $k$ and $G$ , and let $R = k$ . Then $Ho m_{k} (V, W)$ is a $k G$ -module with $g \cdot f (v) = g f (g^{- 1} \cdot v)$ where $g \in G$ and $v \in V$ . It satisfies $(g g^{'}) \cdot f (v) = g g^{'} f (g^{' - 1} g^{- 1} v) = g \cdot (g^{'} f (g^{' - 1} \cdot v))$

If $W = k$ , we call it as a dual representation.