Suppose that we have the following adjoint situation: $$\hom(G-,-)\cong\hom(-,F-),$$ where $F:\cal K\to L$ and $G:\cal L\to K$ are functors.
So if $F$ is a right adjoint then it has a left adjoint $G$. Suppose that the inclusion $F: \cal A\hookrightarrow K$ is right adjoint with reflections maps $r_K:K\to A$ for each $K$ in $\cal K$, so it has left adjoint $G$.
Now, how is $G$ concretely defined for this inclusion $F$?
It's defined by the reflection property of the arrows $r_K$:
For $K\in Ob\mathcal K$, define $G(K):=\mathrm{cod}\, r_K$,
And, for an arrow $f:K\to K'$, there is a unique $\alpha\in\mathcal A$ such that $\alpha\circ r_K=r_{K'}\circ f$.
Define $G(f)=\alpha$, and verify that $G$ is indeed a functor, left adjoint to the inclusion.