I do not understand the role of the empty graph in the definition, numbered (5) in the snippet , of a small-cone-injectivity condition for connected graphs. Is $a\ b$ , the domain of the cone empty ? So in other words what is meant by
presented by the unique morphism from the empty graph ?
A graph is connected if it satisfies two conditions: a) that it is non-empty and b) that any pair of vertices is connected by a directed path.
Categorically, condition b) asserts that any graph morphism $a\ b\to G$ where $a\ b$ is the graph with two vertices and no edges factors through a morphism from $a\ b$ to $a\to\bullet\to\cdots\bullet\to b$. Thus condition b) asserts exactly injectivity with respect to the cone with vertex $a\ b$ and components the morphisms from $a\ b$ to $a\to\bullet\cdots\to\bullet\to b$.
Condition a) can also be expressed as injectivity with respect to a cone. First, by definition a graph is non-empty if and only if it has a morphism from the discrete graph with a single vertex. Since the graphs with morphisms to the discrete graph with a single vertex are precisely the discrete graphs, we have that a graph is non-empty if and only if admits a morphism from a non-empty discrete graph.
Fix a particular non-empty discrete graph. Since every graph has a unique morphism from the empty graph, being non-empty thus reduces to asserting that the morphism from the empty graph factors through the morphism from the empty graph to the fixed non-empty discrete graph. Thus condition a) is injectivity with respect to a cone whose vertex is the empty graph and a single component given by the morphism from the empty graph to a non-empty discrete graph. This is what is mean by a "cone presented by the unique morphism to a non-empty discrete graph".