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I am looking for a reference for the following elementary assertion on complete Segal spaces:

Let $A$ be a bisimplicial set and let $W$ be a complete Segal space. A morphism of $W^A$ is an equivalence if and only if, for each $a\in A_{0,0}$, its image under the map $W^{A}\to W^{\{a\}}\cong W$ is an equivalence of $W$.

I know a proof (which is also valid for Segal spaces), but I suspect that the proof of such an elementary assertion must already be recorded elsewhere. Can anyone locate a reference? Thanks in advance.

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The assertion is stated and proved as Proposition 2.21 in arxiv.2311.01101.

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