What are the morphisms of Adj(C,T) - the category whose objects are the adjunctions of a given monad?
The definition of morphism of adjunctions may be found in MacLane's book. Let $F:\mathcal C\rightarrow\mathcal D$, $G:\mathcal D\rightarrow\mathcal C$, $F':\mathcal C'\rightarrow\mathcal D'$, $G':\mathcal D'\rightarrow\mathcal C'$, and let $(F,G,\eta,\varepsilon)$, $(F',G',\eta',\varepsilon')$ be adjunctions. A map of adjunctions from the first to the second is a pair of functors $L:\mathcal C\rightarrow\mathcal C'$, $K:\mathcal D\rightarrow\mathcal D'$ such that $KF=F'L$, $LG=G'K$ and $L\eta=\eta'L$ (or, equivalently, $\varepsilon K=K\varepsilon'$).

In the case of the category of adjunctions of a monad, the category $\mathcal C$ on which the endofunctor of the monad acts is fixed, so $\mathcal C'=\mathcal C$ and one considers the above definition only in case $L=Id_{\mathcal C}$.