Nonsymmetric monoidal product on $[\mathbb N,\mathbf{Sets}]$

Let $\mathbf P$ be the category with objects the natural numbers and $\hom(m,n)=Sym(n)$ if $n=m$, and the empty set otherwise. It is symmetric monoidal wrt the sum of natural numbers, and has $0$ as a unit object.

Following the lines of this, I can endow $[\mathbf{P},\mathcal V]$, for any cosmos $\cal V$, with a monoidal structure, defining $$ F_1\star\dots\star F_m = \int^{n_1,\dots, n_m}\mathbf{P}(n_1+\dots+n_m,-)\otimes_{\cal V}F_1n_1\otimes_{\cal V}\dots\otimes_{\cal V}F_mn_m $$ Now I can define $F^{\star n}$ to be the $n$-fold convoluted product of $F$ with itself, and I can endow $[\mathbf{P},\mathcal V]$ with another (nonsymmetric) monoidal structure, define $$ T\diamond S := \int^m T(m)\otimes_{\cal V} S^{\star m} $$ My problem is to show the following identity: for any $F\in [\mathbf P,\mathcal V ]$, one has $$ (\heartsuit)\qquad F^{\star n}\cong \mathbf{P}(n,-)\diamond F = \int^m\mathbf{P}(n,m)\otimes_{\cal V}F(m). $$ Is it true? And if it is, how can I show it?

Note: I tried to keep my question self-contained to help anybody understand my request; if it seems to you that I'm implying the real origin of the problem, I'm trying to prove Lemma 3.1 here, and I'ms tuck in the row tagged with "by Yoneda". As far as I can understand, the author is exploiting the identitiy $(\heartsuit)$ to conclude his proof.

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