A Specific Representation of $SO(3)$

Question

I am studying quantum mechanics, particularly angular momentum and rotations at the moment. The notes I am using are K. Schulten's Notes on Quantum Mechanics. In these notes there is a particular representation of $SO(3)$ discussed (on page 104), namely one that acts on elements of $\mathcal C^\infty(\mathbb R^3)$. It does so in the following manner. For any element $R_n(\vartheta)$, where $\vartheta$ is the angle of rotation and $n$ is the axis of rotation, we define the linear map $\rho (R_n(\vartheta))$ via $$\rho(R_n(\vartheta))\psi(r) = \psi(R_n(\vartheta)^{-1}r).$$ The notes claim that this map satisfies the homomorphism property, but upon investigating I end up getting that $$\rho(R_n(\vartheta)R_m(\varphi))\psi(r) = \rho(R_m(\varphi))\rho(R_n(\vartheta))\psi(r)=\psi(R_m(\varphi)^{-1}R_n(\vartheta)^{-1}r).$$ The notes claim that this is equal to $\rho(R_n(\vartheta))\rho(R_m(\varphi))\psi(r)$, but I don't know how to justify this.