ScalingStacks

Construction 2.4.3. Given a map of reduced $\infty$-operads $\mathcal{P}\to\mathcal{Q}$ we get a forgetful functor $G\colon\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)$, such that $U_{\mathcal{P}}G=U_{\mathcal{Q}}$. The unit map

$$\operatorname{Id}\to U_{\mathcal{Q}}F_{\mathcal{Q}}=U_{\mathcal{P}}GF_{\mathcal{Q}}$$

has an adjunct $F_{\mathcal{P}}\to GF_{\mathcal{Q}}$ and by applying $U_{\mathcal{P}}$ we obtain an induced map of the associated monads (as endofunctors of $\mathcal{C}$):

$$\alpha_{G}\colon T_{\mathcal{P}}=U_{\mathcal{P}}F_{\mathcal{P}}\to U_{\mathcal{P}}GF_{\mathcal{Q}}=U_{\mathcal{Q}}F_{\mathcal{Q}}=T_{\mathcal{Q}},$$

which is well defined up to homotopy.