Let $\langle T,\eta, \mu\rangle$ be a monad over a category $C$.
I want to show the associativity in Kleisli category $C_T$.
Suppose that $f_T$ is an arrow in $C_T$ which correspond to $f: X \to T(Y)$ in $C$, and that $g_T \circ_T f_T$ is a composition in $C_T$ which correspond to $\mu_z \circ T(g) \circ f$ where $g: Y \to T(Z)$.
The associativity in $C_T$ is
$h_T \circ_T (g_T \circ_T f_T) = (h_T \circ_T g_T) \circ_T f_T$
where $f: X \to T(Y)$, $g: Y \to T(Z)$, and $h: Z \to T(W)$.
I think that bellow equastions are hold.
$h_T \circ_T (g_T \circ_T f_T) = \mu_W \circ T(h) \circ (\mu_Z \circ T(g) \circ f)$
$(h_T \circ_T g_T) \circ_T f_T = \mu_W \circ T(\mu_W \circ T(h) \circ g) \circ f = \mu_W \circ T(\mu_W) \circ T(T(h)) \circ T(g) \circ f$
So I want to show $T(h) \circ \mu_Z = T(\mu_W) \circ T(T(h))$, but for me it is difficult.