"Define a coproduct $\displaystyle \Delta : L \rightarrow L \otimes L $ by $\displaystyle \Delta(x) = x \otimes 1 + 1 \otimes x $, for some $\displaystyle x \in L $

Show that $\displaystyle \Delta$ defined above is a universal enveloping algebra homomorphism."

Do I simply take $\displaystyle x,y \in L$, and prove that $\displaystyle \Delta(xy) = \Delta(x)\Delta(y) $ ??