Awodey's Category Theory, question 10 of section 2, asks us to show that discrete posets are projective in the category of posets. His proof in the solutions section is rather explicit. Is the following valid? I have already proved that the one-element discrete poset is projective, and that retracts of projective objects are projective; these appear as earlier exercises.

Let $A$ be a discrete nonempty poset, and write $1$ for the discrete poset with one element and one arrow. Define $s: 1 \to A$ which selects some element of $A$, and define $r: A \to 1$ in the obvious way, sending all objects to the sole object of $1$. Then $r \circ s = \text{id}_1$, so $r$ is a retraction; therefore $A$ is a retract of $1$ and so, since $1$ is projective, $A$ is projective.

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    $\begingroup$ btw. there actually exists a correct way to use projectivity of 1 to prove this, but you need a different lemma: coproducts of projective objects are projective $\endgroup$ – user54748 Sep 10 '15 at 17:24

Your argument makes no use of the hypothesis that $A$ is discrete, so you should be quite suspicious of it. In fact what it shows is that $1$ is a retract of $A$, not the other way around.

Here is a more abstract proof strategy you can try. Let $F : C \to D$ be a functor with left adjoint $G$. Show that if $F$ preserves epimorphisms, then $G$ preserves projectives. Now take $C = \text{Pos}, D = \text{Set}$, and $F$ the forgetful functor.


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