  The "linear dual" of a cocomplete linear category $\mathcal C$ is the
category of all cocontinuous linear functors $\mathcal C \to \mathrm{Vect}$. We
study the questions of when a cocomplete linear category is reflexive
(equivalent to its double dual) or dualizable (the pairing with its dual comes
with a corresponding copairing). Our main results are that the category of
comodules for a countable-dimensional coassociative coalgebra is always
reflexive, but (without any dimension hypothesis) dualizable if and only if it
has enough projectives, which rarely happens. Along the way, we prove that the
category $\mathrm{Qcoh}(X)$ of quasi-coherent sheaves on a stack $X$ is not
dualizable if $X$ is the classifying stack of a semisimple algebraic group in
positive characteristic or if $X$ is a scheme containing a closed projective
subscheme of positive dimension, but is dualizable if $X$ is the quotient of an
affine scheme by a virtually linearly reductive group. Finally we prove
tensoriality (a type of Tannakian duality) for affine ind-schemes with
countable indexing poset.
