Eilenberg–Moore category

Remarks

The functor $F^T$ is left adjoint to $U^T$, and $U^T F^T = T$. The unit of the adjunction is $\eta$, and the monad of the adjunction is $(T, \eta, \mu)$.

Example: for the list monad on sets, $\mathcal{C}^T$ is isomorphic to the category of monoids and monoid homomorphisms. A morphism of algebras $f$ satisfies $f(m_1 \cdots m_n) = f(m_1) \cdots f(m_n)$, which is exactly the condition to be a monoid homomorphism.

Non-example: the category whose objects are monoids and whose morphisms are all functions between them is not $\mathcal{C}^T$. A function such as $n \mapsto n + 1$ on $(\N, +)$ does not satisfy clause 2, since it does not send $0 + 0$ to $1 + 1$.

The notation $\mathcal{C}^T$ is standard. The full subcategory of free algebras is equivalent to the Kleisli category of $T$.