Let $\mathcal{M}$ be the simplicial set defined by the formula $Hom( \Delta^{J}, \mathcal{M} ) =Hom( \Delta^{J^{op} } \star \Delta^{J}, \mathcal{C} )$, so that an $n$-simplex of $\mathcal{M}$ is a $(2n+1)$-simplex of $\mathcal{C}$. The inclusions of $\Delta^{J^{op} }$ and $\Delta^{J}$ into $\Delta^{J^{op} } \times \Delta^{J}$ induce a left fibration $\mathcal{M} \rightarrow \mathcal{C}^{op} \times \mathcal{C}$, which is the left fibration you are looking for. For details see section 4.2 of my paper "Derived Algebraic Geometry X", entitled "Twisted Arrow $\infty$-Categories".

Your more general question can be phrased as follows: given a coCartesian fibration $q: X \rightarrow S$ classified by a functor from $S$ into $\infty$-categories, how can you explicitly construct a Cartesian fibration classified by the same functor? First, construct a simplicial set $Y$ with a map $Y \rightarrow S$, such that maps $T \rightarrow Y$ classify maps $T \rightarrow S$ together with maps from $X \times_{S} T$ to the $\infty$-category of spaces. Then $Y \rightarrow S$ is a coCartesian fibration, whose fibers over a vertex $s \in S$ is the $\infty$-category of presheaves on $X_{s}^{op}$. Restricting to representable presheaves determines a full simplicial subset of $Y_0 \subseteq Y$, and the map $Y_0^{op} \rightarrow S^{op}$ is the Cartesian fibration you're looking for.

Let $\mathcal{M}$ be the simplicial set defined by the formula $Hom( \Delta^{J}, \mathcal{M} ) =Hom( \Delta^{J^{op} } \star \Delta^{J}, \mathcal{C} )$, so that an $n$-simplex of $\mathcal{M}$ is a $(2n+1)$-simplex of $\mathcal{C}$. The inclusions of $\Delta^{J^{op} }$ and $\Delta^{J}$ into $\Delta^{J^{op} } \star \Delta^{J}$ induce a left fibration $\mathcal{M} \rightarrow \mathcal{C}^{op} \times \mathcal{C}$, which is the left fibration you are looking for. For details see section 4.2 of my paper "Derived Algebraic Geometry X", entitled "Twisted Arrow $\infty$-Categories".

Your more general question can be phrased as follows: given a coCartesian fibration $q: X \rightarrow S$ classified by a functor from $S$ into $\infty$-categories, how can you explicitly construct a Cartesian fibration classified by the same functor? First, construct a simplicial set $Y$ with a map $Y \rightarrow S$, such that maps $T \rightarrow Y$ classify maps $T \rightarrow S$ together with maps from $X \times_{S} T$ to the $\infty$-category of spaces. Then $Y \rightarrow S$ is a coCartesian fibration, whose fibers over a vertex $s \in S$ is the $\infty$-category of presheaves on $X_{s}^{op}$. Restricting to representable presheaves determines a full simplicial subset of $Y_0 \subseteq Y$, and the map $Y_0^{op} \rightarrow S^{op}$ is the Cartesian fibration you're looking for.