I've seen two different ways to define induced representation.

One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of $H$, then the induced representation $Ind^G_H V$ is the representation of $G$ with $$Ind^G_H V=\{f:G\longrightarrow V|f(hx)=\rho_V(h)f(x)\forall x\in G, h\in H\}$$ and the action $g(f)(x)=f(xg)$, $\forall g\in G$.

If we choose a representative $x_{\sigma}$ from each right $H$-coset $\sigma$ of $G$, then any $f\in Ind^G_H V$ is uniquely determined by $\{f(x_\sigma)\}_{\sigma}$.

I've also seen another way to define induced representation. Let $G$ be a group, $H$ a subgroup of it, and $V$ a representation of $H$. The underlying vector space of $Ind^G_H V$ is the direct sum $$\bigoplus_{\tau\in G/H}\tau V$$ with $\tau$ going over all the left cosets in $G/H$. It multiplication operation is defined by choosing a set $\{g_\tau\}_{\tau\in G/H}$ of coset representatives and setting $$g(\tau v)=\beta (hv)$$ where $\tau v$ is an element in $\tau V$, $\beta$ is the unique left coset containing $gg_{\tau}$, and $gg_{\tau}=g_{\beta}h$ for some $h\in H$. It's easy to verify this definition of $Ind|^G_HV$ does not depend on the choice of the representatives $\{g_\tau\}_{\tau\in G/H}$.

Induced representation is the left adjoint to the restricted representation. So any definition of it should be unique up to unique isomorphism. But I cannot see why the two definitions above are equivalent. Can anybody show me where the problem is? Thanks.