A lax monoidal functor carries monoid objects to monoid objects

Define ${\mu }^{\mathrm{\prime }}:=F(\mu )\circ F_{2;M,M}$ and ${\eta }^{\mathrm{\prime }}:=F(\eta )\circ F_0$. These are the only typed maps with the displayed source and target.

For associativity, paste the lax associativity square for $F$ with the image under $F$ of the monoid-object associativity square for $\mu$. Both composites from $(F(M)⊠F(M))⊠F(M)$ to $F(M)$ equal $F(\mu \circ (1_M\otimes \mu )\circ {\alpha }_{M,M,M})$ with the same inserted $F_2$-maps. Hence ${\mu }^{\mathrm{\prime }}$ is associative.

The left and right unit diagrams for ${\eta }^{\mathrm{\prime }}$ are obtained in the same way by pasting the lax unit squares with the image under $F$ of the two unit diagrams for $(M,\mu ,\eta )$. Thus ${\eta }^{\mathrm{\prime }}$ is a two-sided unit for ${\mu }^{\mathrm{\prime }}$.

Therefore $(F(M),{\mu }^{\mathrm{\prime }},{\eta }^{\mathrm{\prime }})$ is a monoid object in $\mathcal{D}$.