There is only one comonoid object in Set

Question

Let $(X, \eta : X\longrightarrow i, \mu : X\longrightarrow X\sqcup X)$ be an monoid object in Set. Because $i$ is intitial, $i = \emptyset$. But there is only one map $X\longrightarrow\emptyset$ and only if $X = \emptyset$, namely $\eta = \emptyset$.It follows $(X, \eta, \mu) = (\emptyset,\emptyset , \emptyset)$. Is it correct?