TITLE: Pullbacks and pushouts in the category of digraphs?
QUESTION [7 upvotes]: By definition the category of digraphs is:

Objects are endomorphisms of the category $\mathbf{Rel}$ (that is sets equipped with a binary relation on that set).
Morphisms from an object $\mu$ to an object $\nu$ are discretely continuous functions, that is functions $f$ such that $f\circ\mu\subseteq\nu\circ f$. (This formula is equivalent to $\mu\subseteq f^{-1}\circ\nu\circ f$ and to $f\circ\mu\circ f^{-1}\subseteq\nu$.)
Composition is the composition of functions.

I have proved that in categories like this all small products and all small coproducts are defined.
To prove that this category is (co)complete it remains to show existence of pullbacks and pushouts.
Please help me to examine existence of pullbacks and pushouts in the category of digraphs.

REPLY [8 votes]: Since it might be of interest to porton or others, I'll expand on my earlier comment, that the category of digraphs is in fact a Grothendieck quasitopos. Hence it is not only complete and cocomplete, it is also locally cartesian closed, among other things. 
First, recall that the category of 'multigraphs' $\mathbf{Gph}$ (as in Tom Hirschowitz's answer) is the topos of presheaves $G^{op} \to \mathbf{Set}$ on the category 
$$G = (0 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} 1).$$ 
Then, I claim that the inclusion $i: \mathbf{Dig} \to \mathbf{Gph}$ in Tom's answer is the same as the inclusion of separated presheaves for the $\neg\neg$-topology. It is well-known that separated presheaves for a Grothendieck topology form a Grothendieck quasitopos (see for example Johnstone's Elephant). 
The claim is not very hard to check. The representable $\hom_G(-, 1)$ is the digraph $\bullet \to \bullet$, and the representable $\hom_G(-, 0)$ is the digraph $\bullet$; it may be calculated that the only $\neg\neg$-dense subpresheaf of $\hom(-, 0)$ is $\hom(-, 0)$ itself, and that the only $\neg\neg$-dense subpresheaf of $\hom(-, 1)$ besides itself is the digraph inclusion 
$$(\bullet \;\;\; \bullet) \hookrightarrow (\bullet \to \bullet).$$
Now a presheaf $X$ on $G$ is separated if, for each object $c$ and each $\neg\neg$-dense inclusion $i: F \hookrightarrow \hom(-, c)$, the induced map 
$$X(c) \cong \mathbf{Set}^{G^{op}}(\hom(-, c), X) \stackrel{\mathbf{Set}^{G^{op}}(i, X)}{\to} \mathbf{Set}^{G^{op}}(F, X)$$ 
is injective. Since we have only one non-trivial dense inclusion (the digraph inclusion displayed above), the separation condition on $X$ boils down to saying that the canonical map 
$$X(1) \to \hom((\bullet \;\;\; \bullet), X) \cong X(0) \times X(0)$$ 
is injective, which is to say that the map taking each edge $e \in X(1)$ to the source-target pair $(s^\ast(e), t^\ast(e))$ is injective. But this just means $X$ is isomorphic to a digraph (in the sense of this post), so we are done. (Details can be made available on request.)