Let $A,B$ be sets and $\mu:{\cal P}(A)\to {\cal P}(B),$and $\iota:{\cal P}(B) \to {\cal P}(A),$where $\cal P$ denotes the power set. Let $X,X'\subseteq A$ and $Y,Y'\subseteq B.$ Suppose that we have a Galois connection
(i) $X\subseteq X' \Rightarrow \mu(X)\supseteq \mu(X')$
(ii) $Y \subseteq Y' \Rightarrow \iota(Y)\supseteq \mu(Y')$
(iii) $X\subseteq\iota\mu(X)$
(iv) $Y\subseteq\mu\iota(Y)$
Let $T\subseteq A$.
Which inclusion holds in general:
$$\iota\mu(T)\subseteq\bigcup_{a\in T}\iota\mu(\{a\}),$$ or the reverse one:
$$\iota\mu(T)\supseteq\bigcup_{a\in T}\iota\mu(\{a\})$$
? Do (iii) and (iv) follow from (i) and (ii)? Why?
For $a \in T$, you have $\{a\} \subseteq T$, whence $$\mu(\{a\}) \supseteq \mu(T),$$ by (i) and $$\iota\mu(\{a\}) \subseteq \iota\mu(T),$$ by (ii), yielding $$\iota\mu(T) \supseteq \bigcup_{a \in T} \iota\mu(\{a\}).$$