# Universal Enveloping algebra of a L$_\infty$ algebra

In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $$(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$$ from the category of $$A(m)$$-algebras to the category of $$L(m)$$-algebras. Then there is a left adjoint $$\mathcal{U}_L$$for that functor. They define the universal enveloping $$A(m)$$-algebra for a $$L(m)$$-algebra $$L$$ to be $$\mathcal{U}_L(L)$$.

In addition the category of $$A(m)$$-algebras has a strict monoidal structure, which is given by taking the free presentations $$\mathcal{F}_m(X_A)/R_A$$ and $$\mathcal{F}_m(X_B)/R_B$$ for two $$A(m)$$-algebras $$A\equiv(X_A,\left\lbrace \mu_i\right\rbrace),B\equiv(X_B,\left\lbrace \tilde{\mu}_i\right\rbrace)$$ and define $$$$A\Box B\equiv \mathcal{F}_m(X_A\oplus X_B)/(R_A,S_{AB},R_B)$$$$

where $$S_{AB}$$ are the relations generated by $$\sum_\sigma \chi(\sigma) \lambda_i(x_{\sigma(1)},\dots,x_{\sigma(i)})=0$$ ($$\lambda_i$$ are the i-ary operations for $$A\Box B$$). For two $$L(m)$$-algebras $$L_1,L_2$$ it holds that $$$$\mathcal{U}_L(L_1\oplus L_2)\cong \mathcal{U}_L(L_1)\,\Box\, \mathcal{U}_L(L_2)$$$$ and that the homomorphism $$\Delta:\mathcal{U}_L(L)\rightarrow \mathcal{U}_L(L\oplus L)\cong \mathcal{U}_L(L)\,\Box\, \mathcal{U}_L(L)$$ gives the universal enveloping $$A(m)$$ algebra the structure of coassociative cocommutative coalgebra in the category $$(\mathcal{A}(m),\Box,1)$$. Here $$1$$ is the unit $$A\Box 1\cong 1\Box A\cong A$$ obtained from taking the free $$A(m)$$ algebra for the trivial vector space.

My question is, if this is the same as having a $$A_\infty$$-coalgebra structure in the sense of operations $$\left\lbrace \omega^i:X\rightarrow X^{\otimes i}\right\rbrace$$ on the underlying vector space of $$\mathcal{U}_L(L)$$?

If this holds true, are the $$A_\infty$$-algebra and coalgebra structures automatically compatible, i.e. is $$\mathcal{U}_L(L)$$ a $$A_\infty$$-bialgebra?