Understanding of proof Ostrik Theorem 1

Question

I am currently reading the proof of Theorem 1 on page 1o in Victor Ostrik's Module categories (https://arxiv.org/abs/math/0111139).

I don't understand why the functor he defines $N \mapsto \mathrm{Hom}(N, M)$ goes to $\mathcal{C}$ - why is $\mathrm{Hom}(N, M)$ part of $\mathcal{C}$?

Sorry if I am just blind... I'm thankful for any hint :D

Answer 1

In the paper, the functor is not defined by $N\mapsto \operatorname{Hom}(N,M)$, it is actually defined by $N\mapsto \underline{\operatorname{Hom}}(N,M)$. This bifunctor $\underline{\operatorname{Hom}}$ is an internal hom-functor, so it is an object of $\mathcal{C}$ (as in Definition 11).