Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ a smooth representation of $H$. Here smoothness of $\pi$ means that for every $v \in V$ there exists a compact open subgroup $K \le G$ such that $\pi(k)v = v$ for every $k \in K$, and likewise for $\sigma$. Also, we take it that $\pi$ and $\sigma$ are $R$-linear representations where $R$ is a commutative ring; that is, $V$ and $W$ are $R$-modules and the actions are $R$-linear. If it makes a difference, feel free to assume below that $R = \mathbb{C}$.

The smooth induction $\mathrm{Ind}_H^G \sigma$ is the $R$-module of all functions $f\colon G \rightarrow W$ such that $f(hg) = \sigma(h) f(g)$ for all $h \in H$ and $g \in G$ and that $f(gk) = f(g)$ for all $g \in G$ and $k \in K$ for some compact open subgroup $K \le G$ that depends on $f$. The action of $g\in G$ on $\mathrm{Ind}_H^G \sigma$ is by $f \mapsto f(*g)$, so $\mathrm{Ind}_H^G \sigma$ is a smooth $R$-linear representation of $G$. My question is: is the "projection formula map" $$ \pi \otimes_R \mathrm{Ind}_H^G \sigma \rightarrow \mathrm{Ind}_H^G(\pi|_H \otimes_R \sigma) $$ that sends $\sum v_i \otimes f_i$ to $g \mapsto \sum \pi(g)v_i \otimes f_i(g)$ an isomorphism?

I see that it is a well-defined $G$-homomorphism, but I don't see whether it is injective or is surjective. Nor do I see how to write down an inverse.