weight vector. Consequently, since ${\mathbb P}(V({\lambda})) \times {\mathbb P}(V(\mu))$ has a unique closed orbit, in order to prove the claim it suffices to prove that $\psi_{{\lambda},\mu}$ is smooth in $x=([v_{\lambda}],[v_\mu])$ and that the inverse image of $[v_{{\lambda}+\mu}]$ is $x$. The second claim is clear for weight reasons.

In order to prove that $\psi_{{\lambda},\mu}$ is smooth in $x$, consider $T$-stable complements $U \subset {V({{\lambda}})}$, $V \subset {V({\mu})}$ and $W\subset
  {V({{\lambda}+\mu})}$ of ${\Bbbk}\,v_{\lambda}$, ${\Bbbk}\,v_\mu$ and ${\Bbbk}\,v_{{\lambda}+\mu}$. So in a neighbourhood of $x$ the map $\psi_{{\lambda},\mu}$ can be described as $$\psi:U\times V {\longrightarrow}W \;\text{ where } \psi(u,v)= u v_\mu + v_{\lambda}v + u v.$$ The differential of $\psi_{{\lambda},\mu}$ in $x$ is then given by the differential of $\psi$ in $(0,0)$, thus it is described as follows: $$d\psi_x(u,v)=uv_\mu+v_{\lambda}v.$$ Suppose that $d\psi_x$ is not injective. Since it is $T$-equivariant, consider a maximal weight $\eta \in {\Pi({{\lambda}+\mu})}{\smallsetminus}\{{\lambda}+\mu\}$ such that there exists a couple of non-zero $T$-eigenvectors $(u,v)\in \ker d
  \psi_x$ with weights respectively $\eta - \mu$ and $\eta - {\lambda}$. Suppose that $\eta - \mu \in {\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ is not maximal and take ${\alpha}\in {\Delta}$ such that $\eta - \mu + {\alpha}\in
  {\Pi({{\lambda}})}{\smallsetminus}\{{\lambda}\}$ and $e_{\alpha}u\neq 0$: then $$(e_{\alpha}u)v_\mu + v_{\lambda}(e_{\alpha}v) = e_{\alpha}(u v_\mu + v_{\lambda}v) = 0$$ 