I have a puzzle.

Define the category **DiscretePosEnrich** to be the category of \$$\mathcal{V}\$$-enriched categories where \$$\mathcal{V}\$$ is any monoidal preorder over a discrete poset. Morphisms over this category are maps \$$\phi : \mathcal{X} \to \mathcal{Y} \$$ obeying:

\$$\mathcal{X}(a,b) \leq_{\mathcal{X}} \mathcal{X}(c,d) \implies \mathcal{Y}(\phi(a),\phi(b)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \$$

\$$\mathcal{X}(a,b) \otimes_{\mathcal{X}} \mathcal{X}(c,d) \leq_{\mathcal{X}} \mathcal{X}(e,f) \implies \mathcal{Y}(\phi(a),\phi(b)) \otimes_{\mathcal{Y}} \mathcal{Y}(\phi(c),\phi(d)) \leq_{\mathcal{Y}} \mathcal{Y}(\phi(e),\phi(f)) \$$

\$$\mathcal{X}(a,b) = I \implies \mathcal{Y}(\phi(a),\phi(b)) = I \$$

Let **Grp** be the category of [Groups](https://en.wikipedia.org/wiki/Abelian_group) with group homomorphisms as its morphisms.

**Puzzle MD1**. Show that there is mapping \$$T : \mathbf{DiscretePosEnrich} \hookrightarrow \mathbf{Grp}\$$ which is *surjective* up to isomorphism. Lift this map into a functor.