To perform the biblical act of הוצאה, one must do both an “akira” (uprooting an time from its place) and “hanacha” (placing it “down” in the second reshus.

There are two forbidden types of “hotza’ah” (umbrella term for transferring items between private and public domain. One is the explicit issur in the Torah of taking from private domain and placing in public domain. The derivative is called הכנסה״” -transfer from public domain to private domain.

In the scenario of the poor man standing in public domain and rich man inside his home, we have eight scenarios.

The poor man picks picks up and places it inside rich mans hands. (Poor man “hachnasa” and the rich man is permissible.)

Poor one extends his arm, takes item out of the rich man’s hand, and takes it outside. (Poor man “hotzaah” and rich man is permissible)

Rich man picked something up extended his arm and places it in the poor mans hand outside. (Rich man did “hotzzah” and poor man is permissible.)

Rich man extends his arm take something out of the poor mans hands, then brings the item inside. (Rich man has done “hachnasa” and the poor man is permissible) 5. The poor man uprooted an item and extended his hand inside, but the rich man took it out of his hand. (Both are exempt but forbidden, as poor man did akira and rich man did hanacha) 6. The poor man lifted an item and placed it into the outstretched hand of the rich man, who then brought the item into private domain. (Both are exempt, as again, the poor man did akira and the rich man did hanacha) 7. Rich man extends his hands outside and the poor man takes it out of his hand. (Both are exempt, as the rich man did akira and poor man did the hanacha)

The rich man lifted an item and placed it in the outstretched hand of the poor man, who then brought it outside. (Both are exempt, as again, the rich man did akira and the poor man did the hanacha) The mishna describes these cases as 2 that are 4 for “inside” and 2 that are 4 “outside.” (The 2 are hotzaah and hachnasa and the 4 still need to be defined, bc it would seem that there are 4 scenarios for each the rich and poor man doing a rabbinic either Akira or hanacha)

There is another mishna in shavuos where the mishna says yetzios of Shabbos are 2 that are 4. There the mishna means there are 2 explicit in the Torah (hotzaah from poor and rich man) which are 4 (a hachnasa from rich and poor man). There the mishna does not bother discussing the cases where only akira or hanacha are done.