Let $\mathbf{Fcd}$ and $\mathbf{Top}$ be categories (the latter's objects are topological spaces but this doesn't matter for the question). The objects of $\mathbf{Fcd}$ are called endofuncoids (what also does not matter).
Every object $A$ of these categories has so called "underlying set" $\operatorname{Ob}A$.
Morphisms of both categories are of the form $(A,B,f)$, where $A$ and $B$ are objects and $f$ is a function $\operatorname{Ob}A\to\operatorname{Ob}B$.
$F:\mathbf{Top}\to\mathbf{Fcd}$ and $T:\mathbf{Fcd}\to\mathbf{Top}$ are functors which preserve "function" part of the morphisms (that is on morphisms these functors change only objects).
Both $F$ and $T$ do not change underlying sets of objects (that is, for example, $\operatorname{Ob}Fs=\operatorname{Ob}s$).
$F$ happens to be a full and faithful embedding.
Now I want to prove that $T$ is a left adjoint of $F$, with the natural bijection preserving the functions in the morphisms.
Now I will show the scheme which my proof follows. The question is to find a pattern in this proof scheme and provide me with a more general description. (Maybe this is related with monads? If yes, describe my proof in terms of monads, please. Or any other "scheme" my proof fit into?)
We need to prove (abusing in the notation that the morphisms are equal to functions):
$f\in\operatorname{Mor}(T\mu,s) \Leftrightarrow f\in\operatorname{Mor}(\mu,Fs)$ (for every enddofuncoid $\mu$ and topology $s$).
Because $F$ is full and faithful we equivalently transform (this step of the proof is what I want you to generalize) the above to
$$f\in\operatorname{Mor}(FT\mu,Fs) \Leftrightarrow f\in\operatorname{Mor}(\mu,Fs).$$
Then in my proof I prove the last formula.
The full definitions and the proof are available in this online draft, section "Mappings between endofuncoids and topological spaces".