endomorphisms $f,g$ in ${\mathcal{S}}$ and any object $X,Y$ of ${\mathcal{S}}$.

Invertible objects {#sect-invertible-objects}
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An object $I$ of ${\mathcal{S}}$ is *invertible* if there is an object $J$ of ${\mathcal{S}}$ such that $I \otimes J \simeq {\mathbb{1}}$ (or equivalently $J \otimes I \simeq {\mathbb{1}}$ because ${\mathcal{S}}$ is symmetric). Then $J$ is unique (up to isomorphism) and is both a left and right dual of $I$. It follows directly from the definition that the monoidal product of two (and so finitely many) invertible objects is an invertible object. Consequently, the isomorphism classes of invertible objects form a group with the multiplication induced by $\otimes$ and the class of ${\mathbb{1}}$ as the identity element.

Let $I$ be an invertible object of ${\mathcal{S}}$. For all $X,Y \in {\mathrm{Ob}}({\mathcal{S}})$, the map $${\mathrm{Hom}}_{\mathcal{S}}(X,Y) \to {\mathrm{Hom}}_{\mathcal{S}}(X\otimes I, Y \otimes I), \quad f \mapsto f \otimes {\mathrm{id}}_I$$ is bijective, with inverse $${\mathrm{Hom}}_{\mathcal{S}}(X\otimes I, Y \otimes I) \to {\mathrm{Hom}}_{\mathcal{S}}(X,Y), \quad g \mapsto ({\mathrm{id}}_Y \otimes \psi)(g \otimes {\mathrm{id}}_J)({\mathrm{id}}_X \otimes \psi^{-1})$$ where $\psi {\colon}I \otimes J \to {\mathbb{1}}$ is an isomorphism. In particular, the map $${\mathrm{End}}_{\mathcal{S}}({\mathbb{1}}) \to {\mathrm{End}}_{\mathcal{S}}(I), \quad \alpha \mapsto \alpha\, {\mathrm{id}}_I$$ is an isomorphism of monoids (and so ${\mathrm{End}}_{\mathcal{S}}(I)$ is commutative). We denote by ${\left\langle\cdot\right\rangle}_I$ the inverse isomorphism: for 