# In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is self-adjoint, it is unitary.

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An element $a$ of $C^*$-algebra is called an isometry if $a^* a = 1$. This coincides with the usual notion when applied to $B(H)$. Maybe we can just generalize this terminology to the setting of dagger categories. So call a morphism $f$ an isometry if $f^{\dagger} f = 1$. I don't know if this is standard.
This seems to be a good definition. To illustrate, let $A$ denote a matrix with real entries, not necessarily square. Suppose $A^\top A = I.$ Then $$Ax \bullet Ay = x^\top A^\top Ay = x^\top y = x \bullet y.$$ So $A$ preserves dot-products. Hence it preserves lengths: $$\| Ax\| = (Ax \bullet Ax)^{1/2} = (x \bullet x)^{1/2} = \|x\|$$ – goblin Feb 19 at 10:22