Cryptic crossword clue
Setback for old archbishop’s double 4

The most common and most likely answer to this crossword clue is the 4 letter word DUAL

I've seen this crossword clue in The Times
D
U
A
L


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The straight part
We know that "double" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case double = dual.
The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.
This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.

What does DUAL mean?

DUAL noun
  1. Of an item that is one of a pair, the other item in the pair.
  2. Of a regular polyhedron with V vertices and F faces, the regular polyhedron having F vertices and V faces.
  3. (grammar) dual number The grammatical number of a noun marking two of something (as in singular, dual, plural), sometimes referring to two of anything (a couple of, exactly two of), or a chirality-marked pair (as in left and right, as with gloves or shoes) or in some languages as a discourse marker, "between you and me". A few languages display trial number.
  4. Of a vector in an inner product space, the linear functional corresponding to taking the inner product with that vector. The set of all duals is a vector space called the dual space.
DUAL verb
  1. To convert from single to dual; specifically, to convert a single-carriageway road to a dual carriageway.
DUAL adjective
  1. Exhibiting duality; characterized by having two (usually equivalent) components.
  2. Acting as a counterpart.
  3. Double.
  4. (grammar) Pertaining to grammatical number (as in singular and plural), referring to two of something, such as a pair of shoes, in the context of the singular, plural and, in some languages, trial grammatical number.
  5. Being the space of all linear functionals of (some other space).
  6. Being the dual of some other category; containing the same objects but with source and target reversed for all morphisms.

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