The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base

The maps $a_j\circ T(p_j):TL\to U^{T}Dj$ form a cone because every arrow of $D$ is an algebra homomorphism. By [L2] there is a unique $\mathrm{\ell }:TL\to L$ with $p_j\circ \mathrm{\ell }=a_j\circ T(p_j)$ for every $j$.

After composition with every $p_j$, the equations $\mathrm{\ell }\circ {\eta }_L=1_L$ and $\mathrm{\ell }\circ T(\mathrm{\ell })=\mathrm{\ell }\circ {\mu }_L$ become the corresponding algebra laws for $a_j$ by naturality; the limit legs are jointly monic by uniqueness in [L2], so both equations hold and $(L,\mathrm{\ell })$ is a $T$-algebra.

If a cone from an algebra $(X,x)$ has underlying mediating arrow $u:X\to L$, then composing $u\circ x$ and $\mathrm{\ell }\circ T(u)$ with every $p_j$ gives the same map because each cone leg is an algebra homomorphism; joint monicity makes $u$ an algebra homomorphism. Its uniqueness and the uniqueness of $\mathrm{\ell }$ follow from the base limit, giving exactly the strict lift required by [L1], including when $\mathcal{J}$ is empty.