Algebraic formula for co-products in the category of digraphs

Question

I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$.

Morphisms of digraph are functions which preserve $E$.

So we have a category.

It is easy to show that products of $n$ digraphs $F_{i\in n}$ in this category can be defined by the formula:

$\prod F = \bigcap_{i \in n} ( \pi_i^{- 1} \circ F_i \circ \pi_i)$ (where $\pi_i$ are canonical projections).

Question: Can co-products be also described by a simple algebraic formula like above?

Answer 1

$$ \coprod_i (V_i,E_i) = \left(\coprod_i V_i\ ,\ \bigcup_i \nu_i [E_i]\right)$$

Here $\nu_i$ is the inclusion map $V_i \to \coprod_i V_i$, and $\nu_i [-]$ denotes the direct image map on relations induced by it.