I'm reading about Monads on Wiki, I'm confused about these two defining diagrams.
Where $T$ is the endofunctors $\mu: T^2 \to T$ and $\eta: 1_C \to T$. So, my question is, why is the two arrow in left diagram $T \mu $ and on the right diagram $\eta T$? Couldn't we just have $\mu$ only on left and $\eta$ only on right?
No? $\mu$ runs $T^2\implies T$. To get a transformation $T^3\implies T^2$ you need to do something to $\mu$, and the two natural choices that work in general are the 'whiskered' transformations $\mu T$ and $T\mu$, defined via components $\mu_{TX}:T^3X=T^2(TX)\to T^2X$ and $T(\mu_X):T^3X=T(T^2X)\to T(TX)=T^2X$ respectively.
These two transformations are potentially different. The monad coherence diagrams say they agree after applying $\mu$, which is generally useful in computations and diagram chasing.