Definability in the embeddability ordering of finite directed graphs, II

We deal with first-order definability in the embeddability ordering $( \mathcal{D}; \leq)$ of finite directed graphs. A directed graph $G\in \mathcal{D}$ is said to be embeddable into $G' \in \mathcal{D}$ if there exists an injective graph homomorphism $\varphi \colon G \to G'$. We describe the first-order definable relations of $( \mathcal{D}; \leq)$ using the first-order language of an enriched small category of digraphs. The description yields the main result of one of the author's papers as a corollary and a lot more. For example, the set of weakly connected digraphs turns out to be first-order definable in $(\mathcal{D}; \leq)$. Moreover, if we allow the usage of a constant, a particular digraph $A$, in our first-order formulas, then the full second-order language of digraphs becomes available.