Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the Wasserstein distance. Is it possible to define a function \begin{align} \Phi : C( [0,T], \mathcal{P}_2(\mathbb{R})) &\rightarrow C( [0,T], \mathcal{P}_2(\mathbb{R}^2)) \\ \mu \mapsto \pi, \end{align} where $\pi$ satisfies for all $t \in [0,T]$ that $\pi_t \circ pr_2^{-1} = \nu $ and $pr_2$ denotes the projection on the second component.

Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the Wasserstein distance. Is it possible to define a function \begin{align} \Phi : C( [0,T], \mathcal{P}_2(\mathbb{R})) &\rightarrow C( [0,T], \mathcal{P}_2(\mathbb{R}^2)) \\ \mu \mapsto \pi, \end{align} where $\pi$ satisfies for all $t \in [0,T]$ that $\pi_t \circ pr_2^{-1} = \nu_t $ and $pr_2$ denotes the projection on the second component. Edit: It should also hold that $\pi_t \circ pr_1^{-1} = \mu_t$ for all $t \in [0,T].$