In
- Moggi, Eugenio, Computational lambda-calculus and monads (1989): page 2.
A monad over a category $\textbf{C}$ is defined as a triple, $(T, η, µ)$, consisting of the natural transformations $T: \textbf{C} \rightarrow \textbf{C}$, $\eta: Id_{\textbf{C}} \rightarrow T$ and $µ: T^2 \rightarrow T$ for which the following equations hold:
$$(i) \hspace{0.3cm}µ_{TA} ; µ_A = T(µ_A); µ_A$$ $$(ii) \hspace{0.3cm}η_{TA} ;µ_A = Id_{TA} = T(η_{A}); µ_{A} $$
However, I am confused by what the semicolon is denoting in $(i)$ and $(ii)$, and by what the equations mean, since Moggi has not defined what is meant by $µ_{TA}$, $µ_{A}$, $id_{TA}$, etc. Does he simply mean by $µ_{TA}$, the application of $µ$ to $T$ and $A$?
The same mysterious semicolon notation is used in his definition of a Kleisli Triple.
I'm having trouble relating $(i)$ and $(ii)$ to the following 'coherence conditions' (which I take $(i)$ and $(ii)$ to express):
$$(i') \hspace{0.3cm}\mu \circ T\mu = \mu \circ \mu T \hspace{0.4cm}(\text{as natural transformations} \hspace{0.2cm} T^{3} \rightarrow T);$$ $$(ii') \hspace{0.3cm}\mu \circ T \eta = \mu \circ \eta T = 1_{T}\hspace{0.4cm}(\text{as natural transformations} \hspace{0.2cm} T \rightarrow T; \text{here} \hspace{0.2cm} 1_{T} \hspace{0.2cm} \text{denotes the identity transformation from} \hspace{0.2cm} T \hspace{0.2cm} \text{to} \hspace{0.2cm} T).$$
The semicolon here just denotes composition (in reverse order from the usual convention) and the subscripts indicate what object you are evaluating the natural transformations on. So for instance, $\mu_{TA};\mu_A$ means the composition of $\mu_A:T^2A\to TA$ with $\mu_{TA}:T^3A\to T^2 A$ (which would normally be written $\mu_A\circ\mu_{TA}$). With this interpretation, the equations (i) and (ii) are exactly the same as (i') and (ii') except that the natural transformations have been evaluated at a specific object $A$.