We attach a \( \mathfrak{sl}_2 \) crystal, called cocrystal, to a symplectic Kashiwara-Nakashima (KN) tableau, whose vertices are skew KN tableaux connected via the Lecouvey-Sheats symplectic jeu de taquin. These cocrystals contain all the needed information to compute right and left keys of a symplectic KN tableau. Motivated by Willis' direct way of computing type \( A \) right and left keys, we also give a way of computing symplectic, right and left, keys without the use of the symplectic jeu de taquin. On the other hand, we prove that Baker virtualization by folding \( A_{2n−1} \) into \( C_n \) commutes with dilatation of crystals. Thus we may alternatively utilize this Baker virtualization to embed a type \( C_n \) Demazure crystal, its opposite and atoms into \( A_{2n−1} \) ones. The right, respectively left keys of a KN tableau are thereby computed as \( A_{2n−1} \) semistandard tableaux and returned back via reverse Baker embedding to the \( C_n \) crystal as its right respectively left symplectic keys. In particular, Baker embedding also vitualizes the crystal of Lakshmibai-Seshadri paths as \( B_n \)-paths into the crystal of Lakshmibai-Seshadri paths as \( \mathfrak{S}_{2n} \)-paths. Lastly, as an application of our explicit symplectic right and left key maps, thanks to the isomorphism between Lakshmibai-Seshadri path and Kashiwara crystals we use, similarly to the \( Gl_n(\mathbb C) \) case, left and right key maps as a tool to test whether a symplectic KN tableau is standard on a given Schubert or Richardson variety in the flag variety \( Sp_{2n}(\mathbb C)/B \), with \( B \) a Borel subgroup.