The unit object of a multitensor category is semisimple

The Eckmann--Hilton argument makes $E=\mathrm{End}(\mathbf{1})$ a finite-dimensional commutative $k$-algebra. If $a^2=0$, put $J=\mathrm{im}a$ and $K=\ker a$. By [F3], $J\otimes J=0$ and $K\otimes J=0$. Tensoring $0\to K\to \mathbf{1}\to J\to 0$ by $J$ and using [F2] then gives $J=0$, hence $a=0$. A nonzero nilpotent has a nonzero square-zero power, so $E$ is reduced. Thus the commutative Artinian algebra $E$ is a finite product of fields. Its primitive idempotents split $\mathbf{1}$ as a finite direct sum of indecomposable component units ${\mathbf{1}}_i$, each with $\mathrm{End}({\mathbf{1}}_i)$ a field (not necessarily $k$).

Fix a component and a simple subobject $S\subseteq {\mathbf{1}}_i$, which exists by [F1]. Dualizing $0\to S\to {\mathbf{1}}_i\to Q\to 0$ and then tensoring on the left by $S$ gives an exact sequence $0\to S\otimes Q^{\vee }\to S\to S\otimes S^{\vee }\to 0$. The last object is nonzero by the coevaluation zig-zag, so simplicity of $S$ makes $S\to S\otimes S^{\vee }$ an isomorphism.

The coevaluation followed by the inverse of the isomorphism in step 2.1 is a nonzero epimorphism $p:{\mathbf{1}}_i\twoheadrightarrow S$. If $j:S\hookrightarrow {\mathbf{1}}_i$ is the inclusion, then $jp$ is a nonzero element of the field $\mathrm{End}({\mathbf{1}}_i)$, hence an isomorphism. Therefore $j$ is also epic and thus an isomorphism. So every component unit is simple, and step 1.1 together with [F4] makes $\mathbf{1}$ semisimple.