As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Probably this is already known, therefore I would like to know good references for this construction. I would also like to see pictures or a pictorial description of this construction.

Let $\Gamma = (V,E,s,t)$ and $\Gamma' = (V',E',s',t')$ be quivers. Define a new quiver $\Gamma \otimes \Gamma' = (V^{\otimes},E^{\otimes},s^{\otimes},t^{\otimes})$ as follows: The maps $s,t$ correspond to a map $E \times 2 \to V$, which gives a map $E \times 2 \times E' \to V \times E'$. Similarily, $s',t'$ induce $E \times 2 \times E' \to E \times V'$. The set of vertices is the pushout $V^{\otimes}=(V \times E') \cup_{(E \times 2 \times E')} (E \times V')$. The set of edges is $E^{\otimes} =E \times E'$. The obvious map $E^{\otimes} \times 2 \to V^{\otimes}$ yields source and and target maps $s^{\otimes},t^{\otimes} : E^{\otimes} \rightrightarrows V^{\otimes}$.

Thus, an edge in $\Gamma \otimes \Gamma'$ is a pair $(e,e')$ of edges $e: u \to v, e' : u' \to v'$ in $E$ and $E'$, with source $(u,e') \equiv (e,u')$ and target $(v,e') \equiv (e,v')$.

In fact, this endows the category of quivers with a symmetric monoidal structure, compatible with colimits in each variable. The unit is the quiver $\bullet \rightarrow \bullet$.

thinkthe unit should be the null graph/quiver on one vertex.) – Jan Grabowski Jan 10 '13 at 15:42`$\mathrm{Set}$`

-valued functors on the category with two objects and parallel morphisms between them. Then your tensor product seems to be the Day convolution product (see ncatlab.org/nlab/show/Day+convolution) for a particular monoidal structure on this indexing category. – Karol Szumiło Jan 10 '13 at 17:56`$\bullet$`

denotes a quiver with one vertex and no edges, do I read it correctly that`$\bullet \otimes \bullet = \varnothing$`

? If so then your monoidal structure is not a Day convolution in the standard sense since then product of representables would be representable. However, it is a theorem of Day that every closed monoidal structure on a category of (co)presheaves is a Day convolution with respect to somepromonoidalstructure on the indexing category... – Karol Szumiło Jan 11 '13 at 8:23