Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two linear maps $f,g \in Hom(C,A)$ as $f*g:= \mu \circ f\otimes g \circ \Delta$.
Bearing this in mind, Sweedler defines a Hopf algebra as a bialgebra $(H,\mu, \eta,\Delta,\epsilon)$ together with a linear map $S:H\longrightarrow H$ called the antipode of $H$ which satisfies that $S*Id_{H}=Id_{H}*S= \epsilon \circ \mu$, that is, $S$ is an inverse for the identity map with respect to the convolution. Until this point I don't have any existencial crisis.
However, some days ago I attended a conference where the speaker defined a Hopf algebra as a bialgebra $(B,\mu, \eta,\Delta,\epsilon)$ where the following maps are bijective: $(\mu \otimes Id_{B}) \circ (Id_{B} \otimes \Delta)$ and $(Id_{B} \otimes \mu) \circ (\Delta \otimes Id_{B})$. Here is where my existencial crisis comes, why these two definitions are equivalent? It doesn't seem obvious to me how one implies the other, specially, it doesn't seem true that the second implies the first one( I guess the idea behind it is that the bijectivity if these two maps implies the existence of the antipode but again it's not so clear for me how to construct the map $S$ with these hypothesis). As you can see, there is definitively something I'm missing; I can't convince myself yet.
Any hint in order to prove this claim?
In advance thank you very much.