= Solution
{c}
A <monad> is an endofunctor $T:\mathcal C\to\mathcal C$ with <natural transformations> $\eta:1_{\mathcal C}\to T$ and $\mu:T^2\to T$ satisfying
$$
\mu\,T\eta=\mu\,\eta T=1_T,\qquad \mu\,T\mu=\mu\,\mu T.
$$
Its <Kleisli category> has the same objects as $\mathcal C$, with $\mathcal C_T(A,B)=\mathcal C(A,TB)$, identity $\eta_A$, and composite
$$
g\star f=\mu_C T(g)f\qquad(f:A\to TB,\ g:B\to TC).
$$
The unit laws follow from naturality of $\eta$ and the two monad unit laws. For $h:C\to TD$, the two triple composites reduce, using naturality of $\mu$, to
$$
\mu_D\mu_{TD}T^2(h)T(g)f
\quad\text{and}\quad
\mu_DT(\mu_D)T^2(h)T(g)f;
$$
they agree by the monad associativity law. Thus this is a <category>.
Define the <free functor into a Kleisli category> $J$ and the right-hand <functor> $U$ by
$$
JA=A,\quad J(f)=\eta_Bf,\qquad UA=TA,\quad U(f)=\mu_BT(f).
$$
The identity and composition formulas for $J$ are
$$
J(1_A)=\eta_A,\qquad
J(g)\star J(f)=\mu_CT(\eta_C)T(g)\eta_Bf=\eta_Cgf=J(gf).
$$
For $U$, $U(\eta_A)=\mu_AT\eta_A=1_{TA}$, and
$$
U(g\star f)=\mu_CT\mu_CT^2(g)T(f)
=\mu_C\mu_{TC}T^2(g)T(f)
=\mu_CT(g)\mu_BT(f)=U(g)U(f),
$$
where the last step is naturality of $\mu$. Hence both mappings are <functors>. The hom-set identity $\mathcal C_T(JA,B)=\mathcal C(A,UB)$ gives the <adjunction> $J\dashv U$. Its unit is $\eta$, and its counit at $B$ is the Kleisli arrow represented by $1_{TB}:TB\to TB$. Applying $U$ to that counit gives $\mu_B$, so the induced monad is the specified one.
For any $L\dashv R:\mathcal D\to\mathcal C$ inducing this same <monad>, with counit $\varepsilon$, define the <Kleisli comparison functor>
$$
KA=LA,\qquad K(f:A\to TB)=\varepsilon_{LB}L(f).
$$
The triangular identities give $K(\eta_A)=1_{LA}$, and naturality of the counit together with $\mu=R\varepsilon L$ gives $K(g\star f)=K(g)K(f)$. Explicitly, naturality moves $\varepsilon_{LB}$ past $L(g)$, then moves the resulting counit past $\varepsilon_{LC}$, reducing the composite to $\varepsilon_{LC}L(\mu_CT(g)f)$. Thus $K$ is a <functor>, with $KJ=L$ and $RK=U$, and it sends the Kleisli counit to $\varepsilon_L$.
An adjunction morphism here is required to commute with the left and right adjoints and preserve their adjunction structure. Such a morphism must have the above object map; every Kleisli arrow $f$ factors as its counit after $J(f)$, so its arrow map is forced as well. This proves <initiality of the Kleisli adjunction>:
$$
\boxed{J\dashv U\text{ is initial among adjunctions inducing }(T,\eta,\mu).}
$$
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