Every join-(or meet-) homomorphism is an order-homomorphism.
$f$ is a join-homomorphism $\implies f(x\vee y)=f(x)\vee f(y)$. To show that $f$ is order-homomorphism, I need to show $x\le y\implies f(x)\le f(y)$.
Now $x\le y\implies x\wedge y=x$. I now cannot use $f(x\wedge y)=f(x)$ because $f$ is not injective here. I'm stuck here to move from domain to codomain.
Also if I jump from domain to codomain, how do I convert $f(x\wedge y)$ to $f(x\vee y)$. I'm not getting a way to sort this out.
Same thing happens with meet-homomorphism.
P.S: If any one of the proofs is done, then the other can be done using duality. But I want to prove them separately.
Take a join homomorphism $f:A\to B$. Suppose $x,y\in A$ are elements with $x\leq y$, which is equivalent to $x\vee y=y$. Since $f$ is a function, you have $f(x\vee y)=f(y)$, but this is the same as $f(x)\vee f(y)=f(y)$, which is equivalent to $f(x)\leq f(y)$.
Now can you do the same with $\wedge$?