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Let $G=Sp(4,\mathbb{R})$ and let $\pi$ be an irreducible, unitaryrepresentation of $G$ which is cohomological with respect to trivialcoefficients. 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 beingcohomological behaves under Langlands functoriality. We also considerrepresentations which are cohomological with respect to non-trivialcoefficients.

Functorial transfer of Cohomological Representations from $SP(4,\mathbb{R})$ to $GL(5,\mathbb{R})$