I find it helpful to break down the Galois connection requirement into the "if" and the "only if". Eg for \$$f(a) = 2a\$$, I label the left and right adjoints \$$L\$$ and \$$R\$$, respectively, and look at the various conditions \$$\\{C_{implication}\\}\$$:

(least) $$C_{f \leftarrow R}: f(a) \leq b \Leftarrow a \leq R(b) \textrm{ means } R(b) \leq_{\Bbb{R}} \Bigl\lfloor \frac{b}{2} \Bigr\rfloor$$

(upper bound) $$C_{f \to R}: f(a) \leq b \Rightarrow a \leq R(b) \textrm{ means } R(b) \geq_{\Bbb{R}} \Bigl\lfloor \frac{b}{2} \Bigr\rfloor$$

Similarly,

(greatest) $$C_{L \to f}: L(b) \leq a \Rightarrow b \leq f(a) \textrm{ means } L(b) \geq_{\Bbb{R}} \Bigl\lceil \frac{b}{2} \Bigr\rceil$$

(lower bound) $$C_{L \leftarrow f}: L(b) \leq a \Leftarrow b \leq f(a) \textrm{ means } L(b) \leq_{\Bbb{R}} \Bigl\lceil \frac{b}{2} \Bigr\rceil$$

In the vector subspace linear transform example,

$$C_{T \leftarrow R}: T(v) \subseteq w \Leftarrow v \subseteq R(w)$$ means \$$R(w) = R(w \cap Image(T)) \; \forall w \subseteq W\$$, i.e. \$$Image(T)^{\perp}\$$ is in a very loose sense in the "kernel" of \$$R\$$ (\$$R(Image(T)^{\perp}) = R(\Bbb{0})\$$), and \$$R\$$ "at most" gives you the linear inverse on \$$Image(T)\$$,

$$C_{T \to R}: T(v) \subseteq w \Rightarrow v \subseteq R(w)$$ means \$$Ker(T) \subseteq R(w) \; \forall w \subseteq W\$$ and \$$R\$$ "at least" gives you the linear inverse on \$$Image(T)\$$,

$$C_{L \to T}: L(w) \subseteq v \Rightarrow w \subseteq T(v)$$ means \$$w \subseteq Image(T) \; \forall w \subseteq W\$$, i.e. \$$Image(T)^{\perp} = \Bbb{0}\$$ (otherwise \$$L\$$ is not defined), and \$$L\$$ is "at least" the linear inverse on \$$Image(T)\$$, and

$$C_{L \leftarrow T}: L(w) \subseteq v \Leftarrow w \subseteq T(v)$$ means \$$L(w) \cap Ker(T) = \Bbb{0} \; \forall w \subseteq W\$$ and \$$L\$$ is "at most" the linear inverse on \$$Image(T)\$$.

\$$C_{T \leftarrow R}\$$ gives you "least", \$$C_{T \to R}\$$ gives you "upper bound"; together, \$$R\$$ is the "least upper bound"; \$$C_{L \to T}\$$ and \$$C_{L \leftarrow T}\$$ are "greatest", "lower bound", respectively.