Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:

$c(x, y) = 0$ if $x = 0$; $c(x, y) = 1$ otherwise, and

$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.

My guess is that $\varphi$ should be restricted to be continuous.

Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:

$c(x, y) = 0$ if $x = 0$; $c(x, y) = 1$ otherwise, and

$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.

Indeed, $\varphi^c = -1$, while $\psi_\ell \leq -2$ for all $\ell$.

My guess is that $\varphi$ should be restricted to be continuous.

Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:

$c(x, y) = 1$ if $x \neq 0$; $c(x, y) = 0$ otherwise, and

$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.

My guess is that $\varphi$ should be restricted to be continuous.

Are you sure there aren’t additional conditions on $\varphi$? Because otherwise taking $X = \mathbb R$ and $Y$ to be a one point space, the following gives a counterexample:

$c(x, y) = 0$ if $x = 0$; $c(x, y) = 1$ otherwise, and

$\varphi(x) = 0$ if $x = 0$, $\varphi(x)= 2$ otherwise.

My guess is that $\varphi$ should be restricted to be continuous.