re **Puzzle 185**...

According to the [definition](https://forum.azimuthproject.org/discussion/2169/lecture-32-chapter-2-enriched-functors/p1) a \$$\mathcal{V}\$$-functor \$$F : \mathcal{X}\rightarrow \mathcal{Y}\$$ is a function from objects of \$$\mathcal{X}\$$ to objects of \$$\mathcal{Y}\$$ such that \$$\mathcal{X}(x_0, x_1) \leq \mathcal{Y}(F(x_0), F(x_1))\$$ for all \$$\mathcal{X}\$$-objects \$$x_0, x_1\$$.

Applying this in the case \$$\mathcal{V} = \textbf{Cost}\$$, and bearing in mind the reverse ordering on \$$\textbf{Cost}\$$, this says that a \$$\textbf{Cost}\$$-functor is a function between Lawvere metric spaces that "shrinks all journeys", in that the cost of getting from \$$x_0\$$ to \$$x_1\$$ in \$$\mathcal{X}\$$ is always \$$\geq\$$ the cost of getting from \$$F(x_0)\$$ to \$$F(x_1)\$$ in \$$\mathcal{Y}\$$.

OK, what does this mean when \$$\mathcal{X} = C^\text{op}\times D\$$ and \$$\mathcal{Y} = \textbf{Cost}\$$?

It says the cost of getting from \$$(c_0, d_0)\$$ to \$$(c_1, d_1)\$$ is \$$\geq\$$ the cost of getting from \$$\Phi(c_0, d_0)\$$ to \$$\Phi(c_1, d_1)\$$.

ie the cost of getting from \$$c_1\$$ to \$$c_0\$$ plus the cost of getting from \$$d_0\$$ to \$$d_1\$$ is \$$\geq\$$ the maximum of \$$\Phi(c_1, d_1) - \Phi(c_0, d_0)\$$ and zero – thanks @Simon for explaining how \$$\textbf{Cost}\$$ is a \$$\textbf{Cost}\$$-enriched category!

ie \$$C(c_1, c_0) + D(d_0, d_1) \geq \text{max}(\Phi(c_1, d_1) - \Phi(c_0, d_0), 0)\$$

ie \$$\Phi(c_1, d_1) \leq C(c_1, c_0) + \Phi(c_0, d_0) + D(d_0, d_1)\$$

ie flying directly from \$$c_1\$$ to \$$d_1\$$ never costs more than driving from \$$c_1\$$ to \$$c_0\$$, flying from \$$c_0\$$ to \$$d_0\$$, then driving from \$$d_0\$$ to \$$d_1\$$.