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I am now confused with such problem as title goes. To be exact, the problem is

Does there exist a functor from $A:\mathsf{Field}\to \mathsf{Field}$ with a natural transformation from identity functor $\iota: \operatorname{id}\to A$ such that for each $F$, $A(F)$ is the algebraically closure of $F$ through $\iota_F:F\to A(F)$?

It is not easy rather than first glimpse. Let me explain.

Note that, the existence of algebraic closure only ensures that there exist a map from $\operatorname{Obj}(\mathsf{Field})$ to itself. Since the "extension" property is not unique, it is not generally true that we can extend the map to $\operatorname{Mor}(\mathsf{Field})$ for arbitrary choice of algebraic closure.