Let $P_k$ denote the $k$-th permutoassociahedron and define an operad $\Xi$ by $\Xi(k)=V(P_k)$ where the composotion is given by replacement of variables and renaming. An exanmple of replacement of variable and renaming is $$\Gamma((12)3;(12)3,(12)3,(12)3)=(((12)3)((45)6))((78)9)$$ where $\Gamma$ stands for the composition. Then the algebras over the operad $E\Xi$ where $E$ is right adjoint to the set-of-objects functor are precisely the symmetric monoidal categories with strict unit.
Proof: Let $SymMon_*$ denote the category of symmetric monoidal categories with strict unit and strict functors. Let $\mathcal{C}\in Ob SymMon_*$. We shall define an algebra $\gamma:\Xi\to End_\mathcal{C}$. Let $$\gamma(0)(*)(*):=0$$ Let $g\in V(P_k)$ where $k\geq 1$. Define $\gamma(k)(g)(a_1,\cdots,a_k)$ in the obvious way whenever $a_1,\cdots, a_k$ are objects or morphisms in $\mathcal{C}$. We note that if $f,g\in V(P_k)$, then $$\gamma(k)(f)\cong \gamma(k)(g).$$ Define $W:SymmMon_*\to Alg$ by $W(\mathcal{C}):=\gamma$. Let $F:\mathcal{C}\to\mathcal{D}$ be a strict functor between symmetric monoidal categories. Define $W(F):=F$. That is we want to prove that $\gamma(n)(g)\circ F^n=F\circ \gamma(n)(g)$ for all $n\in\mathbb{N}$ and for all $g\in \Xi(n)$. But this follows since $F$ is a strict functor.
Let $\beta:\Xi\to End_\mathcal{C}$ be an algebra. We want to define a symmetric monoidal category on $\mathcal{C}$ with strict unit. Define$$0:=\beta(0)(*)(*),$$ $$a\otimes b:=\beta(2)(12)(a,b)$$ and $$b\otimes a:=\beta(2)(21)(a,b).$$ We note we are forced to have $$\beta(1)(1)(a)=a.$$ The composition and right action forces all appropriate diagrams to commute. We note that $$a\otimes b\cong b\otimes a,$$ $$a\otimes (b\otimes c)\cong (a\otimes b)\otimes c,$$ and $$0\otimes x=x=x\otimes 0.$$ We get the brading when considering the twist morphism from $(a,b)$ to $(b,a)$ to itself. This means $\mathcal{C}$ has a symmetric monoidal structure with strict unit. So, define $\hat{W}(\beta):=\mathcal{C}$. Define $\hat{W}(F):=F$. This is defined by the algebra map formula.
It is clear that these are inverse to each other. This means that the algebras of this operad are precisely symmetric monoidal categories with strict unit.
Is this right?
Also, what is this operad called?