Questions regarding a proof of Nakayama's lemma.

Question

I refer to this proof of Nakayama's lemma.

1. What is $\varphi^n$? Is it $\underbrace{\varphi\circ\varphi\circ\dots\circ\varphi}_{\text{$n$ times}}$?

2. What is $\varphi\delta_{ij}$?

Answer 1

1. Yes, you are right. The multiplication in the endomorphism ring is composition.
2. Well, $\delta_{ij}$ is the Kronecker delta, it is $0$ if $i \neq j$ and $1$ if $i \neq j$. In your situation it means that $\varphi \delta_{ij}$ is $\varphi$ if $i = j$ and the zero endomorphism if $i \neq j$.