Define Modality, prove that double negation is one
Add some theorems about equivalence of equiv and iso in set_cat
Define a variant of [path_sigma] for hprops

In this case, we don't need to provide the second projection.

Also prove that it's an equivalence.

Also move [equiv_path_sigma_hprop] to HProp.v, because it fits there,
and fits in nicely with the proof.
Robustify isequiv_path_sigma_hprop
Merge pull request #352 from JasonGross/ex-7-12

Define Modality, prove that double negation is one
Merge pull request #354 from JasonGross/sigma-hprop

Define a variant of [path_sigma] for hprops.
Merge pull request #357 from JasonGross/set-cat-morphisms

Add some theorems about equivalence of equiv and iso in set_cat
Add two more kinds of duals

The identifiers L and R refer to which ^op is removed/inverted.
Add generalized version of identity functor

Necessary to get between two identical functor categories which use
different instances of funext.
Functorialize yoneda

Alas, I still don't really understand the Yoneda lemma.

This requires _a lot_ of funext munging.  The rule that I've used is one
instance of Funext for every occurrence that appears in the type,
together with one extra one that downward_closed instances of are used
for, e.g., [path_forall] in the middle of the proof.  Perhaps @mattam82
can suggest a way to not need so many instances of [Funext].

Also, the proof is very long.  I'd be happy if anyone could clean it up.

It's also unfortunately rather slow.  I suspect some combination of
proof size, typeclass resolution, and maybe universe constraint
checking, but I'm not entirely sure.  Help speeding things up would also
make me happy.

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10m58.40s | Total                                                    | 8m42.36s
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2m23.81s  | categories/Comma/ProjectionFunctors                      | 1m44.49s
0m55.14s  | categories/Yoneda                                        | 0m08.05s
0m34.69s  | categories/Adjoint/Composition/AssociativityLaw          | 0m33.46s
0m27.57s  | categories/Grothendieck/PseudofunctorToCat               | 0m24.60s
0m26.48s  | categories/ExponentialLaws/Law4/Functors                 | 0m16.69s
0m21.61s  | categories/Comma/InducedFunctors                         | 0m15.64s
0m18.55s  | categories/FunctorCategory/Dual                          | 0m17.77s
0m17.40s  | categories/Adjoint/UnitCounitCoercions                   | 0m17.55s
0m15.39s  | categories/Functor/Pointwise/Properties                  | 0m12.90s
0m14.97s  | categories/Pseudofunctor/Core                            | 0m12.94s
0m12.98s  | categories/Adjoint/Composition/Core                      | 0m12.56s
0m12.70s  | hit/Flattening                                           | 0m12.66s
0m12.32s  | categories/Adjoint/Dual                                  | 0m11.88s
0m11.54s  | categories/Adjoint/Composition/IdentityLaws              | 0m10.56s
0m11.27s  | categories/Pseudofunctor/FromFunctor                     | 0m11.11s
0m09.87s  | categories/ExponentialLaws/Law2/Functors                 | 0m05.56s
0m09.82s  | categories/ExponentialLaws/Law1/Law                      | 0m09.36s
0m07.85s  | categories/Functor/Prod/Universal                        | 0m08.25s
0m07.81s  | categories/GroupoidCategory/Morphisms                    | 0m08.18s
0m07.15s  | categories/UniversalProperties                           | 0m07.28s
0m07.11s  | hit/Connectedness                                        | 0m06.82s
0m06.78s  | categories/Functor/Prod/Functorial                       | 0m02.97s
0m06.78s  | categories/Comma/Dual                                    | 0m07.54s
0m06.59s  | categories/Adjoint/Paths                                 | 0m06.78s
0m06.44s  | categories/Adjoint/UnitCounit                            | 0m06.37s
0m06.31s  | categories/Structure/IdentityPrinciple                   | 0m06.70s
0m05.75s  | categories/Functor/Sum                                   | 0m05.54s
0m05.66s  | categories/Functor/Paths                                 | 0m06.11s
0m05.59s  | categories/Category/Morphisms                            | 0m05.65s
0m05.04s  | EquivalenceVarieties                                     | 0m05.00s
0m04.96s  | categories/DualFunctor                                   | 0m05.27s
0m04.55s  | categories/Category/Sigma/OnMorphisms                    | 0m04.37s
0m04.09s  | categories/Functor/Attributes                            | 0m02.24s
0m03.64s  | categories/ProductLaws                                   | 0m03.60s
0m03.49s  | types/ObjectClassifier                                   | 0m03.44s
0m03.43s  | categories/Structure/Core                                | 0m03.63s
0m03.30s  | categories/Functor/Composition/Functorial                | 0m00.97s
0m03.08s  | types/Sigma                                              | 0m02.94s
0m03.05s  | categories/FunctorCategory/Functorial                    | 0m02.62s
0m02.98s  | categories/Adjoint/UniversalMorphisms                    | 0m02.69s
0m02.79s  | categories/Limits/Core                                   | 0m01.17s
0m02.71s  | categories/Grothendieck/ToSet                            | 0m02.50s
0m02.67s  | categories/NaturalTransformation/Pointwise               | 0m01.33s
0m02.58s  | categories/NaturalTransformation/Isomorphisms            | 0m01.91s
0m02.53s  | Equivalences                                             | 0m02.58s
0m02.22s  | categories/Category/Sum                                  | 0m02.30s
0m02.21s  | categories/FunctorCategory/Morphisms                     | 0m01.70s
0m02.10s  | categories/Category/Sigma/Core                           | 0m02.08s
0m02.05s  | categories/Comma/Core                                    | 0m02.18s
0m01.94s  | PathGroupoids                                            | 0m01.91s
0m01.88s  | categories/ExponentialLaws/Law   N/A                     | 0m01.03s
0m01.86s  | categories/NaturalTransformation/Dual                    | 0m01.86s
0m01.71s  | categories/HomFunctor                                    | 0m01.78s
0m01.68s  | categories/Functor/Pointwise                             | 0m01.05s
0m01.66s  | contrib/HoTTBook                                         | 0m00.66s
0m01.64s  | categories/NaturalTransformation/Paths                   | 0m01.58s
0m01.45s  | categories/ExponentialLaws/Law1/Functors                 | 0m00.97s
0m01.32s  | categories/Category/Prod                                 | 0m01.30s
0m01.31s  | categories/NaturalTransformation/Composition/Laws        | 0m01.31s
0m01.30s  | types/Prod                                               | 0m01.30s
0m01.28s  | categories/Category/Sigma/OnObjects                      | 0m01.26s
0m01.11s  | categories/categories                                    | 0m00.55s
0m01.11s  | contrib/HoTTBookExercises                                | 0m01.01s
0m01.01s  | categories/CategoryOfSections/Core                       | 0m01.06s
0m00.96s  | categories/NaturalTransformation/Prod                    | 0m00.94s
0m00.94s  | hit/Spheres                                              | 0m00.86s
0m00.88s  | types/Paths                                              | 0m00.84s
0m00.85s  | hit/epi                                                  | 0m00.91s
0m00.85s  | categories/Functor/Prod                                  | 0m00.82s
0m00.84s  | categories/SetCategory/Morphisms                         |   N/A
0m00.83s  | categories/Limits/Functors                               | 0m00.42s
0m00.81s  | categories/KanExtensions/Functors                        | 0m00.44s
0m00.79s  | Trunc                                                    | 0m00.72s
0m00.77s  | categories/Functor/Identity                              | 0m00.84s
0m00.76s  | categories/ExponentialLaws/Law3/Functors                 | 0m00.44s
0m00.67s  | categories/InitialTerminalCategory                       | 0m00.66s
0m00.65s  | types/Universe                                           | 0m00.66s
0m00.65s  | categories/Comma/Projection                              | 0m00.56s
0m00.64s  | types/Sum                                                | 0m00.63s
0m00.62s  | hit/quotient                                             | 0m00.65s
0m00.62s  | categories/NaturalTransformation/Composition/Core        | 0m00.71s
0m00.58s  | categories/KanExtensions/Core                            | 0m00.27s
0m00.57s  | categories/NaturalTransformation/Sum                     | 0m00.52s
0m00.57s  | hit/Circle                                               | 0m00.52s
0m00.56s  | Functorish                                               | 0m00.51s
0m00.53s  | hit/Suspension                                           | 0m00.58s
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0m00.52s  | categories/Functor/Composition/Laws                      | 0m00.58s
0m00.48s  | categories/Functor/Functor                               | 0m00.30s
0m00.46s  | HSet                                                     | 0m00.39s
0m00.45s  | Tactics                                                  | 0m00.45s
0m00.43s  | HProp                                                    | 0m00.43s
0m00.43s  | categories/NaturalTransformation/Composition/Functorial  | 0m00.30s
0m00.42s  | types/Forall                                             | 0m00.44s
0m00.41s  | categories/Cat/Core                                      | 0m00.39s
0m00.39s  | categories/NaturalTransformation/Identity                | 0m00.42s
0m00.38s  | categories/Cat/Morphisms                                 | 0m00.32s
0m00.36s  | categories/Functor/Dual                                  | 0m00.38s
0m00.35s  | categories/ExponentialLaws/ExponentialLaws               | 0m00.26s
0m00.34s  | Misc                                                     | 0m00.29s
0m00.34s  | TruncType                                                | 0m00.33s
0m00.34s  | Fibrations                                               | 0m00.35s
0m00.33s  | categories/Grothendieck/Grothendieck                     | 0m00.26s
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0m00.26s  | categories/Grothendieck/ToCat                            | 0m00.22s
0m00.26s  | categories/Category/Objects                              | 0m00.20s
0m00.26s  | types/Arrow                                              | 0m00.27s
0m00.24s  | categories/Functor/Composition/Core                      | 0m00.29s
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0m00.21s  | categories/Utf8                                          | 0m00.21s
0m00.20s  | categories/Functor/Prod/Prod                             | 0m00.17s
0m00.20s  | categories/SetCategory/Functors/Functors                 | 0m00.19s
0m00.20s  | categories/Comma/Comma                                   | 0m00.17s
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0m00.19s  | categories/GroupoidCategory/Core                         | 0m00.18s
0m00.19s  | FunextVarieties                                          | 0m00.16s
0m00.19s  | categories/FunctorCategory/Core                          | 0m00.16s
0m00.18s  | categories/NaturalTransformation/NaturalTransformation   | 0m00.20s
0m00.18s  | hit/iso                                                  | 0m00.21s
0m00.18s  | categories/FunctorCategory/FunctorCategory               | 0m00.15s
0m00.18s  | categories/NatCategory                                   | 0m00.20s
0m00.18s  | categories/Adjoint/Composition/Composition               | 0m00.15s
0m00.18s  | categories/ExponentialLaws/Law1/Law1                     | 0m00.14s
0m00.18s  | categories/NaturalTransformation/Utf8                    | 0m00.13s
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0m00.16s  | Conjugation                                              | 0m00.19s
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0m00.16s  | categories/Cat/Cat                                       | 0m00.17s
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0m00.16s  | HoTT                                                     | 0m00.16s
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0m00.16s  | Overture                                                 | 0m00.16s
0m00.15s  | categories/Limits/Limits                                 | 0m00.18s
0m00.15s  | categories/Category/Univalent                            | 0m00.14s
0m00.15s  | categories/Profunctor/Representable                      | 0m00.16s
0m00.15s  | categories/ExponentialLaws/Law2/Law2                     | 0m00.16s
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0m00.14s  | Contractible                                             | 0m00.12s
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0m00.14s  | categories/ExponentialLaws/Law4/Law4                     | 0m00.09s
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0m00.14s  | categories/Functor/Pointwise/Pointwise                   | 0m00.10s
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0m00.12s  | categories/DiscreteCategory                              | 0m00.10s
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0m00.12s  | types/Unit                                               | 0m00.13s
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0m00.11s  | coq/Init/Peano                                           | 0m00.12s
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0m00.10s  | Pointed                                                  | 0m00.08s
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0m00.10s  | UnivalenceAxiom                                          | 0m00.08s
0m00.10s  | categories/Structure/Utf8                                | 0m00.08s
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0m00.09s  | types/UniverseLevel                                      | 0m00.07s
0m00.09s  | categories/Category/Notations                            | 0m00.11s
0m00.08s  | coq/Init/Specif                                          | 0m00.07s
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0m00.08s  | coq/Init/Logic_Type                                      | 0m00.05s
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