Is monadicity preserved by the underlying functor? Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\mathsf{T}) \to \mathcal{V}$ is tautologically monadic in  $\mathcal{V}$-Cat. If we pass to the underlying categories $\mathsf{U}_0: \mathsf{Alg}(\mathsf{T})_0 \to \mathcal{V}_0$, do we still get a monadic functor?
$\mathsf{U}_0$ is still a right adjoint because $(-)_{0}: \mathcal{V}\text{-Cat} \to \text{Cat}$ is a $2$-functor, but
I expect a negative answer to my question. Yet, I can't find a counterexample.
 A: Following up on my comment, I think the direct proof is easiest - it’s just an exercise in translating classical definitions into the enriched setting. Let’s call the underlying monad $T_0$.
For objects, observe that an algebra of $T$ is given by $a: I \to V(TA, A)$ (which is a map $TA \to A$ in the underlying category) so that $\rho^{-1};(\eta_A \otimes a);m = j_A$ (which is exactly $\eta_A;a = id $ in the underlying category) and $\rho^{-1};((a;T) \otimes a);m= \rho^{-1};(\mu_A \otimes a);m$ (which is precisely $T_0(a);a = \mu_A;a$ in the underlying category).
Let $(A,a), (B,b)$ be algebras if T. A map $f:a \to b$ is a point $f:I \to V(A,B)$ so that $T_0(f)b = af$, that is $f;\rho^{-1};(T \otimes b)m = f;\lambda^{-1};(a \otimes id )m$. This is equivalent to saying $f$ is a point of $eq(\rho^{-1};(T \otimes b)m, \lambda^{-1};(a \otimes id )m) = eq(T;V(TA,b), V(a,B))$ which is the hom-object between the algebras.
A: The answer of Ben MacAdam is very solid and concrete, but I could not believe his result because it did not meet my intuition. In the process of trying to prove him wrong, I managed to have a better understanding of the situation. Indeed Ben is just right.
Excursus. For a monad $\mathsf{T}$ in $\mathcal{V}$-Cat, its category of algebras $\mathsf{Alg}(\mathsf{T})$ is a lax limit. Thus Ben's result shows that the $2$-functor $(-)_0$ preserve a certain kind of $2$-limit. From this point of view, his result is even more ambitious, and I could not believe it. Until...
Yet another proof. When $(\mathcal{V}, \otimes, I)$ is cocomplete, there is an adjunction $ (-)^I:\text{Set} \leftrightarrows \mathcal{V}: \mathcal{V}(I,-) $. Such an adjunction yields a left $(2$-)adjoint for $(-)_0$, $$F: \text{Cat} \leftrightarrows \mathcal{V}\text{-Cat} : (-)_0. $$
This means in particular that $(-)_0$ preserves $2$-limits. As shown by Gray in
The existence and construction of Lax limits, a lax limit is always the $2$-limit of another diagram (and this construction changes the diagram functorially). This proves that $(-)_0$ must preserve Eilenberg-Moore objects.
