It was very helpful when I asked the first part of the proof, so maybe someone can help me with the second one too. I am reading in https://arxiv.org/abs/math/0111139 Theorem 1 on page 10.
The problem I have are in part (4) and (5).
I have the following conditions: Let $\mathcal{C}$ be a semisimple, rigid, monoidal category with finitely many irreducible object ans irreducible unit object. Let $\mathcal{M}$ be a semisimple, indecomposable module category over $\mathcal{C}$. Let A be an algebra in $\mathcal{C}$.
Have $N_1, M \in \mathcal{M}$ and $X, Y \in \mathcal{C}$.
In the proof it says in (4) "It is clear that there exist $X, Y \in \mathcal{C}$ such that there is an exact sequence $$Y \otimes M \rightarrow X \otimes M \rightarrow N_1 \rightarrow 0."$$
I don't see this. Where do the maps come from? How do I know that there is a surjective map just like that?
In 5) there is again an exact sequence. Here it says "We know from Lemma 4 that for any object $L \in A-mod$ there exists an exact sequence $$ Y \otimes A \rightarrow X \otimes A \rightarrow L \rightarrow 0$$ for some $X, Y \in \mathcal{C}$".
Lemma 4 says that: "For any $A-$module $L$ and an object $X \in \mathcal{C}$ we have a canonical isomorphism $$\mathrm{Hom}_A(X \otimes A, L) = \mathrm{Hom}(X, L)."$$
Could someone maybe explain why those exact sequences exist? :D That would be wonderful!