A homomorphism of quivers is just a natural transformation of the corresponding functors, #m/def/cat
i.e. a pair of functions and mapping vertices and edges respectively such that
for all , or equivalently
for all .
Proof these are equivalent
Suppose ^H1 holds.
Let , i.e. for some such that and ,
in which case
and thus .
Therefore .
Now suppose ^H2 holds.
Let , , and .
Then , so ,
whence