I don't see where $i$ is used in $C=\blacktriangleright\smallfrown O\smallfrown \sqcup^j$. Perhaps, it is supposed to be written as $C=\blacktriangleright\smallfrown\sqcup^i \smallfrown O\smallfrown \sqcup^j$. Maybe add note that $\sqcup^n$ is a sequence of $n$ $\sqcup$s.
I don't see where $i$ is used in $C=\blacktriangleright\smallfrown O\smallfrown \sqcup^j$. Perhaps, it is supposed to be written as $C=\blacktriangleright\smallfrown\sqcup^i \smallfrown O\smallfrown \sqcup^j$. Maybe add note that $\sqcup^n$ is a sequence of $n$ $\sqcup$s.