In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of strict polynomial functors.
Namely, let $k$ be a commutative ring with unity. Fix $d \in \mathbb{N}$. The $k$-linear category $\Gamma^d \mathrm{P}_k$ is defined as follows: Its objects are finitely generated projective $k$-modules, and the morphism space $\operatorname{Hom}_{\Gamma^d \mathrm{P}_k} \left(V, W\right)$ between two objects $V$ and $W$ is defined to be $\Gamma^d \operatorname{Hom}_k \left(V, W\right)$. Here, for any $k$-module $U$, we let $\Gamma^d U$ be the $S_d$-invariant part of the tensor power $U^{\otimes d}$; when $U$ is finitely generated projective, then $\Gamma^d U$ is also isomorphic to the $k$-module $\left(\operatorname{Sym}^d\left(U^\vee\right)\right)^\vee$ (where $^\vee$ means dual $k$-module), and to the $d$-th graded component of the free divided power algebra on the $k$-module $U$. (The motivation behind this definition of the morphism space is that the elements of $\Gamma^d \operatorname{Hom}_k \left(V, W\right)$ are something like degree-$d$ polynomial maps from $V$ to $W$, in that they can be applied to vectors in $V$ and the result depends on the input vector like a degree-$d$ homogeneous polynomial. To apply a morphism $f \in \Gamma^d \operatorname{Hom}_k \left(V, W\right)$ to a vector $v \in V$, just assume WLOG that $f = f_1 \otimes f_2 \otimes \cdots \otimes f_d$ (otherwise, use linearity), and then the result is $f_1\left(v\right) f_2\left(v\right) \cdots f_d\left(v\right)$. But this is just a motivation, and in general a morphism $f \in \Gamma^d \operatorname{Hom}_k \left(V, W\right)$ is not actually determined by what it does to the vectors $v \in V$.)
The category $\Gamma^d \mathrm{P}_k$ is known as the category of $d$-th divided powers. The $k$-linear functors from $\Gamma^d \mathrm{P}_k$ to the category $\mathrm{M}_k$ of all $k$-modules are known as the strict polynomial functors. (This is a formalization of the concept of a "polynomial functor" from the category of "nice" $k$-modules with $k$-linear maps to the category of $k$-modules with degree-$d$ polynomial maps.)
The internal tensor product $\otimes_{\Gamma^d_k}$ on the category of strict polynomial functors is defined in Proposition 2.4 of Krause's above-mentioned paper. When $k$ is a field of characteristic $0$, this internal tensor product is a categorification of the internal product (a.k.a. inner product, a.k.a. Kronecker product) on the ring of symmetric functions. (More precisely: When $k$ is a field of characteristic $0$, the category $\Gamma^d \mathrm{P}_k$ is semisimple, and its simple objects are the Schur functors. On the Grothendieck group of this category, the internal tensor products of the Schur functors correspond to the internal products of the corresponding Schur functions. Thus, its structure constants are the famous Kronecker coefficients.)
Question. Is there a way to define the internal tensor product of strict polynomial functors more-or-less explicitly (without resorting to Day convolution, as Krause does)?
Notice that this is doable in the case when $k$ is a field of characteristic $0$, because in this case, Schur-Weyl duality allows us to switch between strict polynomial functors and representations of $S_d$ (and on the latter, the internal tensor product corresponds to plain tensor product). Thus, if $F$ and $G$ are two strict polynomial functors over a field $k$ of characteristic $0$, then the internal tensor product of $F$ and $G$ is $T^d \otimes_{k S_d} \left(V \otimes W\right)$, where $V = \operatorname{Hom}\left(T^d, F\right)$ and $W = \operatorname{Hom}\left(T^d, G\right)$. (Fine print: $T^d$ denotes the strict polynomial functor sending each $V$ to $V^{\otimes d}$. For any strict polynomial functor $X$, the morphism space $\operatorname{Hom}\left(T^d, X\right)$ is an $S_d$-module by permuting the tensorands of $T^d$. The tensor product $V \otimes W$ is a tensor product of group representations of $S_d$.)
There are few sources on the internal tensor product in the general case. One is a thesis (Rebecca Reischuk, The monoidal structure on strict polynomial functors); but it also uses the definition by Day convolution.
