  Orthogonal Calculus, first developed by Weiss in 1991, provides a calculus of
functors for functors from real inner product spaces to spaces. Many of the
functors to which Orthogonal Calculus has been applied since carry an
additional lax symmetric monoidal structure which has so far been ignored. For
instance, the functor $V \mapsto \text{BO}(V)$ admits maps $$\text{BO}(V)
\times \text{BO}(W) \to \text{BO}(V \oplus W)$$ which determine a lax symmetric
monoidal structure.
  Our first main result, Corollary 4$.$2$.$0$.$2, states that the Taylor
approximations of a lax symmetric monoidal functor are themselves lax symmetric
monoidal. We also study the derivative spectra of lax symmetric monoidal
functors, and prove in Corollary 5$.$4$.$0$.$1 that they admit
$O(n)$-equivariant structure maps of the form $$\Theta^nF \otimes \Theta^nF \to
D_{O(n)} \otimes \Theta^nF$$ where $D_{O(n)} \simeq S^{\text{Ad}_n}$ is the
Klein-Spivak dualising spectrum of the topological group $O(n)$.
  As our proof methods are largely abstract and $\infty$-categorical, we also
formulate Orthogonal Calculus in that language before proving our results.
