In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$? In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ f^\dagger \circ f \leq f$ (see, for example, here). I have been calling this latter condition _rectangular_ on the basis of the answer I recieved here. However, I am interested in allegory-like structures where this equivalence breaks down, so a name for this condition not equal to 'rectangular' would be great.

Based on my previous answer, what about partial isometries?