After introducing the notion of a strict \(D\)-category, we seek ways to relate different \(D\)-categories and their interleavings to each other. To this end let \(\mathbf{C}\) and \(\mathbf{C}'\) be strict \(D\)-categories with smoothing functor \(\mathcal{S}\) respectively \(\mathcal{S}'\).

Now let \(F \colon \mathbf{C} \rightarrow \mathbf{C}'\) be a \(1\)-homomorphism and let \(A\) and \(B\) be objects of \(\mathbf{C}\).

  1. Corollary. We have \(M_{\mathcal{S}'} (F(A), F(B)) \leq M_{\mathcal{S}} (A, B)\) and \(\mu_{\mathcal{S}'} (F(A), F(B)) \leq \mu_{\mathcal{S}} (A, B)\).

The previous two lemmata have the following

We also note that we can compose \(1\)-homomorphisms. To this end let \(\mathbf{C}''\) be another strict \(D\)-category.

Now we defined \(1\)-homomorphisms as functors with special properties. For any two functors \(F\) and \(G\) from \(\mathbf{C}\) to \(\mathbf{C}'\) we have the class of natural transformations from \(F\) to \(G\). Now suppose \(F\) and \(G\) are \(1\)-homomorphisms, then \(F\) and \(G\) are in some sense compatible with \(\mathcal{S}\) and \(\mathcal{S}'\). We name a natural transformation from \(F\) to \(G\), that is compatible with \(\mathcal{S}\) and \(\mathcal{S}'\), a \(2\)-homomorphism from \(F\) to \(G\).

Just like we can compose natural transformations we can also compose \(2\)-homomorphisms.

Moreover we can compose \(1\)-homomorphisms with \(2\)-homomorphisms and vice versa.