We construct explicit inverse maps between these Hom-spaces.

Step 1: Map $\Phi: \text{Hom}_G(\text{Ind}_H^G(W), V) \to \text{Hom}_H(W, V)$

Given $\phi: \text{Ind}_H^G(W) \to V$ (a $G$-homomorphism), define $\Phi(\phi): W \to V$ by evaluation: $\Phi(\phi)(w) = \phi(\tilde{w})$ where $\tilde{w}: G \to W$ is the function defined by: $\tilde{w}(g) = \begin{cases} w & \text{if } g \in H \\ 0 & \text{if } g \notin H \end{cases}$

More precisely, $\tilde{w}$ satisfies $\tilde{w}(hg) = \rho_W(h)(\tilde{w}(g))$ for $h \in H$, so $\tilde{w} \in \text{Ind}_H^G(W)$.

Verification that $\Phi(\phi)$ is $H$-equivariant: For $h \in H$ and $w \in W$: $\Phi(\phi)(h \cdot w) = \phi(\widetilde{h \cdot w}) = \phi(\rho_H(h) \circ \tilde{w})$

Since $\phi$ is $G$-equivariant and using properties of induced representations: $= \rho_V(h)(\phi(\tilde{w})) = h \cdot \Phi(\phi)(w)$

Step 2: Map $\Psi: \text{Hom}_H(W, V) \to \text{Hom}_G(\text{Ind}_H^G(W), V)$

Given $\psi: W \to V$ (an $H$-homomorphism), define $\Psi(\psi): \text{Ind}_H^G(W) \to V$ by: $\Psi(\psi)(f) = \sum_{i=1}^n \rho_V(g_i) \psi(f(g_i))$ where $\{g_1, \ldots, g_n\}$ is a set of left coset representatives for $H$ in $G$, so $G = \bigsqcup_{i=1}^n g_i H$.

Verification that $\Psi(\psi)$ is well-defined: We need to check independence of coset representatives. If $g_i' = g_i h_i$ for some $h_i \in H$, then: $f(g_i') = f(g_i h_i) = \rho_W(g_i)^{-1} \rho_W(h_i)^{-1} f(e)$ and the formula remains consistent using $H$-equivariance of $\psi$.

Verification that $\Psi(\psi)$ is $G$-equivariant: For $g \in G$: $\Psi(\psi)(\rho_G(g) f) = \Psi(\psi)(f(\cdot g))$ $= \sum_{i=1}^n \rho_V(g_i) \psi(f(g_i g))$

Reindexing and using that multiplication by $g$ permutes cosets: $= \rho_V(g) \sum_{i=1}^n \rho_V(g'_i) \psi(f(g'_i)) = \rho_V(g) \Psi(\psi)(f)$

Step 3: Verify $\Phi$ and $\Psi$ are inverse

$\Phi \circ \Psi = \text{id}$: For $\psi: W \to V$ and $w \in W$: $(\Phi \circ \Psi)(\psi)(w) = \Phi(\Psi(\psi))(w) = \Psi(\psi)(\tilde{w})$ $= \sum_{i=1}^n \rho_V(g_i) \psi(\tilde{w}(g_i))$

Since $\tilde{w}(g_i) = 0$ except when $g_i \in H$ (which we can take as $g_1 = e$): $= \rho_V(e) \psi(w) = \psi(w)$

$\Psi \circ \Phi = \text{id}$: For $\phi: \text{Ind}_H^G(W) \to V$ and $f \in \text{Ind}_H^G(W)$: $(\Psi \circ \Phi)(\phi)(f) = \Psi(\Phi(\phi))(f) = \sum_{i=1}^n \rho_V(g_i) \Phi(\phi)(f(g_i))$ $= \sum_{i=1}^n \rho_V(g_i) \phi(\widetilde{f(g_i)})$

Using properties of $\phi$ and the definition of $\text{Ind}_H^G(W)$, this equals $\phi(f)$.