Duality in Monoidal Categories

"The tensor-hom adjunction is a tool of fundamental importance in representation-theory. In case of finite-dimensional representations of a Hopf algebra with invertible antipode or more generally a tensor category, it manifests in a particularly well-behaved way: every object $X$ has a dual $X^*$ together with evaluation and coevaluation morphisms that give rise to a variant of dual bases expressed in terms of snake identities. These identities readily imply that the functors $X\otimes-$ and $X^*\otimes -$ are adjoint to each other. It is natural to ask whether the converse holds as well: is a monoidal category rigid provided that for every object $X$ there exists a "dual" $DX$ and $X\otimes -$ is adjoint to $DX\otimes -$? We are going to answer this in the negative by studying $sl_2$-crystals. This talk is based on joint work with T. Zorman: https://arxiv.org/abs/2301.03545