I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the link). The author mentions the following result in Algebraic group theory without proof.
Lemma 2.4. Let $G_1,...,G_n$ be linear algebraic groups and $H\subset G_1\times...\times G_n$ be an algebraic subgroup such that all the natural projections $\pi_i:H\to G_i$ are surjective. Then the induced projections between their connected components $\pi_i:H^o\to G_i^o$ are all surjective.
I don't know how to prove this. The lemma follows if the following question is affirmative.
Question. Let $f:H\to G$ be a surjective morphism between algebraic groups. Is it true that the induced map between connected components $f:H^o\to G^o$ is also surjective?
It is known that this is true under certain conditions (e.g., $\mathrm{ker}(f)$ is connected). However, I don't think it is true in general. I have a counter-example for (abstract) topological groups (take $\mathrm{id}:\mathbb R\to \mathbb R$ the identity map between the (discrete) group $\mathbb R$ and the (euclidean) group $\mathbb R$). Still, I cannot find a counter-example for algebraic groups (because of the finiteness of $H/H^o$).
So do you know any proof/reference or idea of how to prove the lemma and the question above?