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Stream: learning: questions

Topic: Codiscrete cospans in V-Cat

David Kern (Jul 29 2024 at 00:49):

In Fibrations in bicategories, Street states as 6.14 that a cospan of $\mathsf{V}$-categories $\mathsf{A}\xrightarrow{u}\mathsf{S}\xleftarrow{v}\mathsf{B}$ ($\mathsf{V}$ a nice monoidal category) is (bi)codiscrete if and only if "every object of $\mathsf{S}$ is isomorphic either to an object of the form $ua$ or an object of the form $vb$" and asserts that this is an easy consequence of the definitions. However, I do not see how this follows so directly, so what obvious argument am I missing?