Let $(T,\mu, \eta)$ be a monad over the category $\textbf A$ , let $(A,a)$ be a $T$- algebra and $m: B\rightarrow A$ be monic. Prove a morphism $b:TB\rightarrow B$ is the structure for an algebra such that $m$ is a morphism of $T$-algebras if and only if the following diagram commutes:
I just don't know where to start. Of course I have to assume the diagram commutes and then see it is a morphism of $T$-algebras and vice-versa , but I get stuck on all the specifics.