Kleisli category

Remarks

The monad laws $\mu \circ T \eta = \mathrm{id} = \mu \circ \eta T$ and $\mu \circ T \mu = \mu \circ \mu T$ give the identity laws and the associativity of $\circ_T$. The functor $\mathcal{C} \to \mathcal{C}_T$, $f \mapsto \eta \circ f$, has a right adjoint $\mathcal{C}_T \to \mathcal{C}$ that sends $X$ to $T X$ and $f$ to $\mu \circ T f$, and this adjunction induces $T$.

Example: let $T$ be the monad $X \mapsto X \sqcup \set{\ast}$ on $\mathbf{Set}$, with $\eta_X$ the inclusion and $\mu_X$ sending both added points to $\ast$. A Kleisli morphism $X \to Y \sqcup \set{\ast}$ is a partial function from $X$ to $Y$, undefined where the value is $\ast$, and $\circ_T$ is composition of partial functions. So $\mathbf{Set}_T$ is isomorphic to the category of sets and partial functions.

Non-example: composing $f \colon X \to T Y$ and $g \colon Y \to T Z$ by $T g \circ f$ alone gives a morphism $X \to T T Z$, not $X \to T Z$. Without the multiplication $\mu$, clause 3 does not produce a morphism of the right type.

For the monad of a variety on $\mathbf{Set}$, the Kleisli category is equivalent to the category of free algebras of the variety and the homomorphisms between them.