Decomposition of powersets

Question

Since every set can be written as the disjoint union of finite subsets, I wonder:

Can the power-set $P(A)$ of a set $A$ be written as a limit/colimit of the powersets $P(A_i)$ of the finite subsets $A_i$ of $A$?

Answer 1

For $F\supseteq G$ finite subsets of $A$, let $m_{F, G}$ denote the map from $P(F)$ to $P(G)$ given by $$m_{F, G}(X)=X\cap G.$$ Unless I'm missing something, the limit of the resulting diagram is exactly $P(A)$.