Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$. I read a textbook on Quantum Groups by Kassel, and did not understand the following proof of well-definedness of $\Delta$ (comult.) and $\epsilon$ (counit) on $GL_q(2)$ and $SL_q(2)$: (pg 84)
The proof is sketchy and describes that to prove well-definedness on $SL_q(2)$, it suffices to show 
$$\Delta (det_q -1)=(det_q -1)\otimes det_q + 1\otimes (det_q -1)$$
(how?)
and $$\epsilon (det_q -1)=0$$
The part I don't understand is how to get the above two equations and how does that prove well-definedness.

Additional theorems that may help which are defined earlier are 
$\Delta (det_q)=det_q \otimes det_q$ and $\epsilon (det_q)=1$.
Sincere thanks for any help.
 A: First, note that $SL_q(2)$ remains an algebra, because $M_q(2)$ is an algebra, and imposing the condition that $\text{det}_q  = 1$ is the same as factoring out by the ideal generated by all multiples of $\text{det}_q -1$ (denoted by $I=(\text{det}_q-1)$.  That it remains a coalgebra follows from the fact that $I$ is also a coideal, that is
$$\Delta(I) \subset SL_q(2) \otimes I + I \otimes SL_q(2)$$
We see that this is true from Kassel's computation of $\Delta(\text{det}_q -1)$.  To get it, he just does the "add a copy, subtract a copy" trick.  You should already know that $\Delta(\text{det}_q) = \text{det}_q \otimes \text{det}_q$.  From the fact that the coproduct is linear you then have
$$ \Delta(\text{det}_q -1) = \text{det}_q \otimes \text{det}_q - 1 \otimes 1 $$
Doing the "adding and subtract a copy trick" we have
$$ \Delta(\text{det}_q) = \text{det}_q \otimes \text{det}_q - 1 \otimes \text{det}_q + 1 \otimes \text{det}_q - 1 \otimes 1$$
Then grouping like tensor products gives us Kassel's computation.
Since the coproduct was an algebra morphism before we passed to the quotient algebra, it remains an algebra morphism afterwards.  You can verify this by writing down the appropriate commutative diagrams for $M_q(2)$ and $SL_q(2)$ and then linking them up with the canonical projection of $M_q(2)$ to its quotient.  (If you need to verify this, I'd suggest you do it generally, with an any hopf algebra $H$ and quotient $H/I$ for a co/ideal $I$).
