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.