Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together.

When defining the connected sum of surfaces with boundary, is the boundary of each surface allowed to touch its removed disk?

By my intuition it seems like a bad idea to allow that, because that might lead to “not nice” behaviours. But I can’t really think of anything in the formal definition that could forbid it – removing an open disk doesn’t prevent the disk from touching the surface boundary.

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