I asked this a week ago at [MSE][1], but nobody replied. Could anybody enlighten me if the following is true? 

Let $F$ and $G$ be [Hopf algebras][2] over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces over $k$). Let 
$$
\varphi:F\to G
$$
be a morphism of Hopf algebras (i.e. $\varphi$ preserves multiplication, comultiplication, unit, counit and antipode), and 
$$
H\subseteq G
$$
a Hopf subalgebra in $G$ (i.e. $H$ is a Hopf algebra with respect to the multiplication, comultiplication, unit, counit and antipode induced from $G$).

Question:

> Is it true that the preimage of $H$,
$$
\varphi^{-1}(H)=\{x\in F:\ \varphi(x)\in H\}
$$
is a Hopf subalgebra in $F$?



  [1]: https://math.stackexchange.com/questions/4754318/is-the-preimage-of-a-hopf-subalgebra-a-hopf-subalgebra
  [2]: https://en.wikipedia.org/wiki/Hopf_algebra