Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $\>2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_H$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_G\times {\mathrm{Bun}}_H$ yields theta-lifting functors $F_G:\mathrm{D}\left({\mathrm{Bun}}_H\right)\to \mathrm{D}\left({\mathrm{Bun}}_G\right)$ and $F_H:\mathrm{D}\left({\mathrm{Bun}}_G\right)\to \mathrm{D}\left({\mathrm{Bun}}_H\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case)....