What equivalence relation are you using to define motives?
The finest that people normally consider is rational equivalence, in which case
two morphisms whose graphs are rationally equivalent (which is to say, roughly speaking, two morphisms which can be deformed one into the other via a family parameterized by $\mathbb P^1$) will give the same morphism.
A trivial, but concrete, example: all constant morphisms from any variety to
$\mathbb P^1$ will be identified in the category of motives.
If you use a coarser relation (homological equivalence or numerical equivalence, which are conjecturally the same, and lead to what are called Grothendieck motives) than even more morphisms of varieties will be identified in the category of motives.
Incidentally, these functors are certainly not full, since one takes a Karoubian closure as part of the construction of the category of motives, and then furthermore inverts the Lefschetz motive.