  Let $G=Sp(4,\mathbb{R})$ and let $\pi$ be an irreducible, unitary
representation of $G$ which is cohomological with respect to trivial
coefficients. Using the inclusion from $SO(5,\mathbb{C})$ to
$GL(5,\mathbb{C})$, we transfer $\pi$ to an irreducible representation
$\iota(\pi)$ of $GL(5,\mathbb{R})$ and determine how the property of being
cohomological behaves under Langlands functoriality. We also consider
representations which are cohomological with respect to non-trivial
coefficients.
