Some questions about the morphisms Given $A,B,C$ three objects and $h,g\in Mor(A,B)$. Suppose to any $u\in Mor(C,A)$, we have $h\circ u=g\circ u$, can we say $g=h$? Moreover, suppose that $p\in Mor(A,A)$, suppose that to any $u\in Mor(C,A)$, we have $p\circ u=u$, can we say $p=Id_A$?

No. For the simplest example: what if the set of morphisms from $C$ to $A$ is empty?