I recently came across an operad called the swiss-cheese operad. I am trying to understand its definition through partial composition. I am having trouble checking the axioms (This makes me think that my understanding of composition is not correct), I would like to write the partial composition the way I understand them.
$\circ_i : S(m,n) \times S(m_1,n_1) \longrightarrow S(m+m_1-1, n+n_1 )$ and $\circ_i : S(m,n) \times D(k) \longrightarrow S(m, n+k-1)$
The first composition is defined by gluing $S(m_1,n_1)$ to the $i^{th}$ semidisk of $S(m,n)$, and the second compostion is defined by gluing the disk $D(k)$ into the $i^{th}$ disk of $S(m,n)$ and erasing the seam.
Note that the labeling runs independently on semi disks and disks. i.e. the labeling on semi-disk in the first composition $1-2-3 \circ_2 \textbf{1-2} = 1-\textbf{2-3}-4$, the labeling on disk in the first composition $1-2 \circ_2 \textbf{1-2} = 1-\textbf{2-3}-4$
The following picture depicts the compositions. $\circ_2 : S(3,2) \times S(2,2) \longrightarrow S(4, 4 )$ and $\circ_2 : S(3,2) \times D(3) \longrightarrow S(3, 4)$
Question: Is this the correct way to compose objects of the Swiss-Cheese operad? If not can someone please explain the correct way? Also, the correct way to label the disks and semi-disks?
Notation: In $S(m,n)$, $m$ denotes the number of semi-disks, and $n$ denotes the number of disks. I avoid drawing disks, instead give numbering.