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I understand why in the category of sets two parallel morphisms $f, g: A \rightarrow B$ are identical iff for each element $x: 1 \rightarrow A$ it holds that $f\circ x = g \circ x$.

Awodey on p. 36 of Category Theory asks (as an exercise), why in any category two parallel morphisms $f, g: A \rightarrow B$ are identical iff for each generalized element $x: X \rightarrow A$ it holds that $f\circ x = g \circ x$.

Could someone please give me a hint how to prove this?

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Just let $X=A$ and $x$ be the identity morphism.

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    The article by "[Doing Without Diagrams](http://www.maths.ed.ac.uk/~tl/elements.pdf)" by Tom Leinster on the other hand seems to confirm the above answer...2015-07-10