This answer does not meet any of the questions, yet it provides the reason for which $i_A$ has a left adjoint. The whole story is contained in **Categories of continuous functors I** by *Kelly* and *Freyd*. Nowadays maybe we have more sophisticated ways of phrasing this result, but the core of the reason is still in the paper by Freyd and Kelly, which offers also a historical tour of all the partial results that led to the theorem.

Coming to your *second question*, it is very hard in general to provide an explicit formula. Since the left adjoint $L_A$ has to coincide by abstract nonsense with $\mathsf{ran}_{i_A}(1),$ there exist a somewhat obscure integral expression coming from (co)end calculus.