ScalingStacks

Let \(f \colon a \rightarrow b\) be a morphism of \(\mathcal O\)-algebras in \(\mathcal P\). Assume that for all \(n \geq 0\) and colors \(X_1, \ldots, X_n, Y\) in \(\mathcal O\), the map of spaces \[\mathrm{Mul}_{\mathcal P}(b_{X_1}, \ldots, b_{X_{n}}; c_{Y}) \xrightarrow{-\circ(f,\ldots, f)} \mathrm{Mul}_{\mathcal P}(a_{X_1}, \ldots, a_{X_{n}}; c_{Y})\] is an equivalence. Then, for any \(m \geq 0\), the map of multi-hom spaces \[\mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{b,\ldots, b}_m; c) \xrightarrow{- \circ (f,\ldots, f)} \mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{a, \ldots, a}_m; c)\] is an equivalence for all \(n\).