nLab
rigid monoidal category

A rigid monoidal category is essentially a monoidal category where every object has a dual. Conventions differ as to the precise way in which this is used.

One convention is as follows: a right dual of an object $V$ in a monoidal category is an object $V^{*}$ equipped with unit $\eta :1\to V^{*}\otimes V$ and counit maps $ϵ:V\otimes V^{*}\to 1$ satisfying the triangle identities (the snake diagrams). Similarly for left duals.

This convention fits in with the standardized conventions regarding adjoint functors. Namely according to this convention, an endofunctor $F:C\to C$ has a right adjoint $F^{*}:C\to C$ if and only if $F^{*}$ is a right dual of $F$ in the monoidal category $\mathrm{End}(C)$.

Definition

A monoidal category is rigid if every object has both a right dual and a left dual.

Note that this definition only asserts the existence of the dual objects. It does not assert that specific duals have been chosen. Nor does it assert that the right dual of an object is isomorphic to its left dual: this need not be the case in general, though it is true in a braided monoidal category, and thus automatically also in a symmetric monoidal category. In practice, algebraic geometers are the most frequent users of the term ‘rigid’, and they focus on the symmetric monoidal case, so they ignore the difference between right and left duals.