Products of ∞ operads

Observation 14.1.2. The \(\infty \)-category \(\Op _{\infty }\) admits finite products. The terminal object is \(\Comm \), and the product of \(\infty \)-operads \(\Oo \) and \(\Pp \) has total category \[ (\Oo \times \Pp )^{\otimes } = \Oo ^{\otimes }\times _{\Span (\Fin )}\Pp ^{\otimes }. \] Indeed, finite products, the decomposition of the fibers, and the required cocartesian lifts are all computed componentwise in this fiber product.