Whenever the endofunctor category exists, monads on a fixed category and their morphisms form a category

The identity $1_T:T\Rightarrow T$ satisfies $1_T{\eta }^T={\eta }^T$ and $1_T{\mu }^T={\mu }^{T}T{1}_T(1_T)T$, so it is a monad morphism.

If $\alpha :T\Rightarrow S$ and $\beta :S\Rightarrow R$ are monad morphisms, then $(\beta \alpha ){\eta }^T=\beta {\eta }^S={\eta }^R$, so their vertical composite preserves the unit.

Naturality of $\beta$ gives $\beta S\circ S\alpha =R\alpha \circ \beta T$; substituting the multiplication equations for $\alpha$ and $\beta$ yields $(\beta \alpha ){\mu }^T={\mu }^{R}R(\beta \alpha )(\beta \alpha )T$, so the composite preserves multiplication.

By [L2], vertical composition is associative and the transformations in step 1.1 are identities. Steps 2.1 and 3.1 give closure under composition under the stated endofunctor-category size condition. Hence these objects and arrows form a category.