For $X_{\bullet,\bullet}$ a bisimplicial set, its diagonal is the simplicial set that is the precomposition with the diagonalfunctor$(Id, Id) \colon \Delta^{op} \to \Delta^{op} \times \Delta^{op}$ on the opposite of the simplex category, i.e. the simplicial set with components:

Danny Stevenson, pages 2 & 11 of: Décalage and Kan’s simplicial loop group functor, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 (arXiv:1112.0474, tac:26-28)

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