$D\to~T-Alg$ is injective on objects

Question

Let $D$ be a monad. I’m trying to prove that the Kleisli category $D_T$ is equivalent to the full subcategory of T-Alg whose objects are all $\mu_X:T^2X\to TX$ for $X\in Ob(D)$. Call the latter category E. There is a fairly obvious functor $D_T\to E$ sending X to $\mu_X:T^2X\to TX$, and $f:X \to TY$ to $\mu_Y \circ T(f)$. This is obviously surjective on objects and I can show it's bijective on hom sets. But I have trouble proving it is injective on objects. Basically can you recover $X$ from $\mu_X:T^2X\to TX$?