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I'm currently learning about Galois connections, and we have to proof some identities. However this is my first abstract math course, and I'm having some trouble with finding the proofs. Any help is appreciated, specifically any tips on proving when a function is a morphism.

a morphism is defined as $a \leq p => f(a) \leq f(p)$

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    What is a Galois connection ? Got a reference ? – 2017-02-19
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    Maybe better to post a specific problem, than to just ask for general advice. – 2017-02-19

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We want to show that $f(a) \leq f(p)$ given that $a \leq p$.

Since $((X, \leq), (Y, \leq), f, g)$ is a Galois connection, this is equivalent to showing that $a \leq g(f(p))$. Since we're assuming that $a \leq p$, it is enough to show that it is always true that $p \leq g(f(p))$.

Now since $((X, \leq), (Y, \leq), f, g)$ is a Galois connection, the statement

$$p \leq g(f(p))$$

is equivalent to

$$ f(p) \leq f(p) $$

which is obviously true since preorders are reflexive.