Etymology
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let (n.)

"stoppage, obstruction" (obsolete unless in legal contracts), late 12c., from archaic verb letten "to hinder," from Old English lettan "hinder, delay, impede," etymologically "make late," from Proto-Germanic *latjan (source also of Old Saxon lettian "to hinder," Old Norse letja "to hold back," Old High German lezzen "to stop, check," Gothic latjan "to hinder, make late"), related to *lata-, source of late (adj.), from PIE root *‌‌lē- "to let go, slacken."

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fly (v.2)

"run away," Old English fleon, flion "fly from, avoid, escape;" essentially a variant spelling of flee (q.v.). In Old English, this verb and fleogan "soar through the air with wings" (modern fly (v.1)) differed only in their present tense forms and often were confused, then as now. In some Middle English dialects they seem to have merged completely. Distinguished from one another since 14c. in the past tense: flew for fly (v.1), fled for fly (v.2).

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-let 

diminutive noun-forming element, Middle English, from Old French -elet, which often is a double-diminutive. It consists of Old French diminutive -et, -ette (see -et) added to nouns in -el, which in many cases represents Latin diminutive -ellus; see -el (2)). "The formation did not become common until the 18th c." [OED].

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let (v.)

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense might be "let go through weariness, neglect."

The idea of slackening lies at the root of both applications of the term. When we speak of letting one go, letting him do something, we conceive him as previously restrained by a band, the loosening or slackening of which will permit the execution of the act in question. ... At other times the slackness is attributed to the agent himself, when let acquires the sense of being slack in action, delaying or omitting to do. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859] 

He points to similar developments in French laisser "to let" from Latin laxare "to slacken," German lassen "to permit, to let," from dialectal lass "loose."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to."

To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.

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fly (n.)

Old English fleoge "a fly, winged insect," from Proto-Germanic *fleugon "flying insect" (source also of Old Saxon fleiga, Old Norse fluga, Middle Dutch vlieghe, Dutch vlieg, Old High German flioga, German Fliege "fly"); literally "the flying (insect)" (compare Old English fleogende "flying"), from PIE root *pleu- "to flow," which is also the source of fly (v.1).

Originally any winged insect (moths, gnats, beetles, locusts, hence butterfly, etc.) and long used by farmers and gardeners for any insect parasite. Flies figuratively for "large numbers" of anything is from 1590s. Plural flien (as in oxen, etc.) gradually normalized 13c.-15c. to -s. Fly in the ointment is from Eccles. x:1. Fly on the wall "unseen observer" first recorded 1881. No flies on _____ "no lack of activity or alertness on the part of," is attested by 1866. Meaning "fish-hook dressed to resemble an insect" is from 1580s; Fly-fishing is from 1650s. Fly-catcher "bird which eats insects on the wing" is from 1670s. The fly agaric mushroom (1788) so called because it was used as a poison for flies.

The sense of "a flight, flying" is from mid-15c. From the verb and the notion of "flapping as a wing does" comes the noun sense of "tent flap" (1810), which was extended to "strip of material sewn into a garment as a covering for buttons" or some other purpose (1844). Baseball fly ball attested by 1866. To do something on the fly is 1856, apparently from baseball.

When the catcher sees several fielders running to catch a ball, he should name the one he thinks surest to take it, when the others should not strive to catch the ball on the fly, but only, in case of its being missed, take it on the bound. ["The American Boys Book of Sports and Games," New York, 1864]
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fly (v.1)

"to soar through air; move through the air with wings," Old English fleogan "to fly, take flight, rise into the air" (class II strong verb; past tense fleag, past participle flogen), from Proto-Germanic *fleugan "to fly" (source also of Old Saxon fliogan, Old Frisian fliaga, Middle Dutch vlieghen, Dutch vliegen, Old High German fliogan, German fliegen, Old Norse flügja), from PIE *pleuk-, extended form of root *pleu- "to flow."

Meaning "go at full speed" is from c. 1300. In reference to flags, 1650s. Transitive sense "cause to move or float in air" (as a flag, kite, etc.) is from 1739; sense of "convey through the air" ("Fly Me to the Moon") is from 1864. Related: Flew; flied (baseball); flown; flying. Slang phrase fly off the handle "lose one's cool" dates from 1825.

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fly (adj.)

slang, "clever, alert, wide awake," by 1811, perhaps from fly (n.) on the notion of the insect being hard to catch. Other theories, however, trace it to fledge or flash. Slang use in 1990s might be a revival or a reinvention.

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let-down (n.)

also letdown, "a disappointment," 1768, from let (v.) + down (adv.). The verbal phrase is from mid-12c. in a literal sense "cause to be lowered," of drawbridges, etc.; figuratively by 1754.

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let-up (n.)

"cessation, restraint, relaxation, intermission," 1837, from verbal phrase let up "cease, stop" (1787). In Old English the phrase meant "to put ashore" (let out meant "put to sea"). Bartlett (1848) says the noun is "an expression borrowed from pugilists."

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shoo-fly (interj.)

admonition to a pest, by 1866 (in political journalism), from shoo (v.) + fly (n.). Popularized by a Dan Bryant minstrel song ("Shoo fly — don't bother me") that appeared in print by 1868 and was enormously popular from the next year, which launched it as a catch-phrase that, according to H.L. Mencken, "afflicted the American people for at least two years" and made an appearance in the Congressional Record.

[A]n epidemic called the "Shoo fly," that has raged in New York and other eastern cities for some time, has at length broken out there [Cincinnati] with great intensity. Young and old have got it. Babies lisp it, and old men are heard to pipe it forth. The child, when threatened with chastisement by its mother, modifies the parental wrath, and sometimes escapes the impending rod by waving its infantile hand and singing out:"Shoo-fly, don't bodder me!"
[Knoxville, Tenn., Daily Press and Herald, Jan. 28, 1870]

 The phrase also was used in various technical senses. The Pennsylvania Dutch delicacy shoo-fly pie is attested by 1908 in a recipe in the Philadelphia Inquirer.

Its name is generally taken to be an allusion to the fact that it is so attractive to flies that they have to be constantly shooed away from it, but the fact that it originated as a Pennsylvania-Dutch specialty suggests the possibility that shoofly is an alteration of an unidentified German word. [Ayto, "Diner's Dictionary"]

Beam's "Pennsylvania German Dictionary" [Lancaster, 1985] gives der Melassichriwwelboi as the translation of "shoofly pie."

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