why this condition implies uniqueness

Question

Suppose that in a category if a morphism $g$ has the same domain as $p$ and $$pu=pv$$ implies $$gu=gv$$ then $$g=tp$$ for some $t$.

Then $t$ is unique. Is it true; how can I (dis)prove this ?

See below:

EDIT for anyone who wants look up the old french papers.

Answer 1

If there are no restrictions on your category then it's easy to see that $t$ is not necessarily unique by constructing a small counterexample.

Consider for example a category of four objects and the following morphisms, plus the four identity morphisms which I haven't drawn:

Notice that there is no ambiguity as to what is the composition of any two composable morphisms.

Notice also that you could have many morphisms $A \to B$ instead of just one; it would still be a counterexample.