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From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...

In the formalism of Lawvere metric spaces, we have that the distance in the hom-space $[X,Y]$ is given by: $$ d(f,g) = \sup_{x\in X} d(f(x),g(x)) . $$ Therefore, a sequence of functions $f_n:X\to Y$ ...