Let $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$ be lattices (in fact in the example I have in mind, they are distributive and even co-Heyting lattices). Let maps $f:\mathfrak{A}\rightarrow\mathfrak{B}$ and $g:\mathfrak{B}\rightarrow\mathfrak{C}$.
Let also $f$ preserve finite (including nullary) joins and $g$ preserve finite (including nullary) meets.
What can be said about the composition $g\circ f$? Specifically, is every map $z:\mathfrak{A}\rightarrow\mathfrak{C}$ representable as a composition $g\circ f$? If not, what are the necessary/sufficient conditions on $z$ to be representable in this way?