= Solution
Use $q\in k^\times$ and the <quantum plane> convention $yx=qxy$. In this convention $SL_q(2)$ is the <coordinate Hopf algebra of quantum SL2> with the catalog parameter replaced by $q^{-1}$. It is generated by $a,b,c,d$, subject to
$$
\begin{gathered}
ba=qab,\quad ca=qac,\quad db=qbd,\quad dc=qcd,\quad bc=cb,\\
da-ad=(q-q^{-1})bc,\qquad ad-q^{-1}bc=1.
\end{gathered}
$$
Equivalently the last determinant relation is $da-qbc=1$. The <quantum determinant> convention is essential: these relations use the inverse of the parameter in the catalog's $ab=qba$ convention.
Writing $T=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$, the <comultiplication> is matrix multiplication, $\Delta t_{ij}=\sum_kt_{ik}\otimes t_{kj}$. Explicitly,
$$
\begin{aligned}
\Delta a&=a\otimes a+b\otimes c,&\Delta b&=a\otimes b+b\otimes d,\\
\Delta c&=c\otimes a+d\otimes c,&\Delta d&=c\otimes b+d\otimes d.
\end{aligned}
$$
The <counit> is $\varepsilon(a)=\varepsilon(d)=1$, $\varepsilon(b)=\varepsilon(c)=0$. Extend $\Delta$ and $\varepsilon$ as unital <algebra homomorphisms over a field>. Matrix multiplication is associative, giving coassociativity; multiplication by the identity matrix gives the <counit> laws. The <antipode> is
$$
\boxed{S(T)=\begin{pmatrix}d&-qb\\-q^{-1}c&a\end{pmatrix}.}
$$
For example $(TS(T))_{11}=ad-q^{-1}bc=1$ and $(S(T)T)_{11}=da-qbc=1$, while the off-diagonal entries vanish by the commutation relations. Thus this is a <Hopf algebra>, not merely an algebra presentation.
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