In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, for a presheaf of $\infty$-groupoids, one needs to apply a transfinite iteration. However, not much detail is given about this. Does anyone know where I can read more about this? Thanks.
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Thomas Nikolaus and Christoph Schweigert discuss the +-construction for $n=2$ in their paper Equivariance in Higher Geometry. They split it up into two steps (I think): first producing a pre-2-stack out of a presheaf of 2-groupoids, and then making it a 2-stack.
Applied to the pre-2-stack obtained by delooping the monoidal stack of principal $U(1)$-bundles, one gets exactly the definition of a bundle gerbe.