Assume $\mathbb A$ is category. Let $h:A \rightarrow A$ be an arrow in $\mathbb A$. I want to know if $h$ is the identity on $A$. Is this true if $\forall X \in obj(\mathbb A) \forall f:A \rightarrow X \in hom(A,X) : f \circ h = f$ ?
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You probably meant $f \circ h = f$, not $f \circ h = h$. And the answer is yes: if that condition holds, taking $f = id_A$ you get $id_A \circ h = id_A$, so $h = id_A$.