  We provide an explicit combinatorial realization of all simple and injective
(hence, and projective) modules in the category of bounded
$\mathfrak{sp}(2n)$-modules. This realization is defined via a natural tableaux
correspondence between spinor-type modules of $\mathfrak{so}(2n)$ and
oscillator-type modules of $\mathfrak{sp}(2n)$. In particular, we show that, in
contrast with the $A$-type case, the generic and bounded
$\mathfrak{sp}(2n)$-modules admit an analog of the Gelfand-Graev continuation
from finite-dimensional representations.
