Let $D$ be a complete and cocomplete symmetric monoidal closed category (a Bénabou cosmos). Let $P$ be the permutation category. Consider the substitution product $\circ$ on the functor category $[P^{op},D].$ Consider the subcategory of $[P^{op},D]$ consisting of $\circ$-monoids and $\circ$-monoid morphisms. Max Kelly claims on page 2 that this category is complete but not necessarily cocomplete.
I know that the category $[P^{op},D]$ is complete and cocomplete. But what about the subcategory of $\circ$-monoids? Are there some general theorems one can apply?
For any monoidal category $\mathscr M$, one can form the category of monoids $\mathbf{Mon}(\mathscr M)$ in $\mathscr M$, which is equipped with a forgetful functor $U : \mathbf{Mon}(\mathscr M) \to \mathscr M$. (Note that the category $\mathbf{Mon}(\mathscr M)$ is not generally a subcategory of $\mathscr M$, because the objects of $\mathscr M$ may have more than one distinct monoid structure.) The forgetful functor $U$ creates limits: explicitly, this means that it preserves and reflects limits, but more conceptually it means that limits in $\mathbf{Mon}(\mathscr M)$ are formed by taking the underlying limits in $\mathscr M$. Thus, if $\mathscr M$ is complete, then so is $\mathbf{Mon}(\mathscr M)$. This applies in particular to the given example of the substitution monoidal structure.
Note, however, that $U$ does not create colimits; more sophisticated conditions are required to ensure that $\mathbf{Mon}(\mathscr M)$ has colimits (see, for instance, Kelly's paper A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on).