I'm taught that, for arbitrary wavefunctions $\psi, \phi$, that:
$$\hat B = |\psi \rangle \langle \phi \mid$$
Which produces a new ket apparently when applied to a ket, as..
$$\hat B \ | \mu \rangle = |\psi \rangle \langle \phi \mid \mu \rangle = C | \psi \rangle$$
Which, following this logic makes sense following it, but how do I know that applying $| \mu \rangle$ to $\hat B$ that it naturally goes where it did other than the fact that "it just fits that way" (the way my brain would reply to this in my head) because bras and kets fit together. But that doesn't sound like a very rigorous understanding on my part. Why does it go to the bra vector and form an inner product other than "it just would" (which is what I get the urge to think)?