# What are the morphisms of Adj(C,T) - the category whose objects are the adjunctions of a given monad?

The Wikipedia page for Monad says just that for a monad $(T,\eta,\mu)$ we can define the category of all adjunctions that define the monad:

Let ￼$\textbf{Adj}(C,T)$ be the category whose objects are the adjunctions ￼$(F,G,e,\varepsilon)$ such that $(GF,e,G\varepsilon F)=(T,\eta,\mu)$￼ and whose arrows are the morphisms of adjunctions which are the identity on ￼$C$. Then this category has

• an initial object $(F_T,G_T,\eta,\mu_T) : C\to C_T$￼, where $C_T$￼ is the Kleisli category,
• a terminal object $(F^T,G^T,\eta,\mu^T) : C\to C^T$￼, where ￼$C^T$ is the Eilenberg-Moore category.

What are the morphisms of adjunctions in $\textbf{Adj}(C,T)$ and what is meant by ... which are the identity on $C$?

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 $L=Id_{\mathcal C}$.
• I'm a bit disturbed by the $=$ sign between functors: surely this means naturally isomorphic? – cody Apr 3 '15 at 17:20