<p>We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal hom is tensor-representable? We provide a counterexample in terms of the category of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathfrak {sl}_2(\mathbb {C})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">sl</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-crystals. As a byproduct, we obtain characterisations of the Grothendieck–Verdier duality and rigidity of functor categories endowed with Day convolution as their tensor product. This has various applications, three of which we study in detail: generalisations of quasi-Frobenius algebras, called QF-2 algebras; Mackey functors, where we prove that, as expected due to work of Bouc, an object being rigidly dualisable is equivalent to it being finitely-generated projective; and crossed modules of finite groups, where we associate to each of these objects a Grothendieck–Verdier category of group-graded representations.</p>