alternate definition of a decreasingly filtered commutative monoid in $\mathfrak{S}$ using properties of functors between symmetric monoidal categories.

\[def:DFO\] A *[decreasingly filtered commutative monoid]{} in $\mathfrak{S}$ is a lax symmetric monoidal functor $$I : (\mathbb{N}^{\op},+,0) \lra (\mathfrak{S},\wedge ,S).$$*

We will write $I_n$ for the functor evaluated on the object $n\in\mathbb{N}^{\op}$. We equip the category of functors $\mathfrak{S}^{\mathbb{N}^{\op}}$ with the projective model structure, where fibrations are defined to be natural transformations $I\rightarrow J$ such that $I_i\rightarrow J_i$ are fibrations for each $i$ and weak equivalences are defined to be natural transformations $I\rightarrow J$ such that $I_i\rightarrow J_i$ is a weak equivalence for each $i$. If $I$ is a cofibrant object in the projective model structure on $\mathfrak{S}^{\mathbb{N}^{\op}}$ then it is a sequence $$\xymatrix{ ... \ar[r]^{f_3} & I_2 \ar[r]^{f_2} & I_1 \ar[r]^{f_1} & I_0 } ,$$ such that each $I_i$ is cofibrant as an object in $\mathfrak{S}$ and the maps $f_i$ are cofibrations for each $i$ [@thhmay].

Due to B. Day [@Day Ex. 3.2.2], the full subcategory of lax symmetric monoidal functors in $\mathfrak{S}^{\mathbb{N}^{\op}}$ is equivalent to the category of commutative monoids in $\mathfrak{S}^{\mathbb{N}^{\op}}$ with the Day convolution symmetric monoidal product. (See [@MR3427193] for a modern treatment of this in the setting 