Decomposing Frobenius Heisenberg categories

We give two alternate presentations of the Frobenius Heisenberg category, ${\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus \cdots \oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ${\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F,k}$ are given in terms of planar diagrams consisting of strands “colored” by integers $i=1,\dots ,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i\mathrm{.}$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, ${\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F_1,k}\otimes \cdots \otimes {\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of ${\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F,k}$ that we call the $\mathit{\text{partial}}$ Karoubi envelope of ${\mathcal{H}\mathcal{e}\mathcal{i}\mathcal{s}}_{F,k}$.