I have the following situation: $\mathcal{M}$ is a semi simple, indecomposable module category over a semisimple, rigid monoidal category $\mathcal{C}$ with finitely many irreducible objects and irreducible unit object. Let $N$ be an object in $\mathcal{M}$.
In my exercise it says that $\mathrm{Hom}(-, N)$ is left exact under those conditions. Why is that?
An equivalent characterization of $\mathcal{M}$ being semisimple is that every $N\in \mathcal{M}$ is injective (see e.g. R is semisimple if and only if every R module is projective for the dual statement). $\operatorname{Hom}(-,N)$ being exact is equivalent to $N$ being injective.